Finite Methods in Mathematical Practice - University of Leeds

Finite Methods in Mathematical Practice Laura Crosilla∗ and Peter Schuster†

Abstract In the present contribution we look at the legacy of Hilbert’s programme in some recent developments in mathematics. Hilbert’s ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so–called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert’s programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid on the use of finite methods. The main aim is to eliminate the ideal objects and in so doing obtain more elementary and informative proofs. We survey some work in commutative algebra—mainly about and around the Zariski spectrum and the Krull dimension of a commutative ring—which witnesses the feasibility of such a revised Hilbert’s programme.

Contents 1 Introduction

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2 Hilbert’s programme now and then 2.1 The original Hilbert’s programme . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Generalised and relativised Hilbert’s programmes . . . . . . . . . . . . . . . . 2.3 Contemporary perspectives of Hilbert’s programme . . . . . . . . . . . . . . .

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3 Finite methods for constructive algebra

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4 Geometric formulas and dynamical proofs

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5 Realising Hilbert’s programme in commutative algebra 5.1 Rings and ideals . . . . . . . . . . . . . . . . . . . . . . . 5.2 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spectra and schemes as distributive lattices . . . . . . . . 5.3.1 Spectral spaces . . . . . . . . . . . . . . . . . . . . 5.3.2 Joyal’s lattice . . . . . . . . . . . . . . . . . . . . . 5.3.3 Radical ideals . . . . . . . . . . . . . . . . . . . . . 5.3.4 Generalisations . . . . . . . . . . . . . . . . . . . . 5.4 Krull dimension of rings and lattices . . . . . . . . . . . . 5.5 Concrete applications of Krull dimension . . . . . . . . . . ∗

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School of Mathematics, University of Leeds, Woodhouse Lane, LS2 9JT, Leeds, England; [email protected]. The author gratefully acknowledges EPSRC grant EP/G029520/1. † School of Mathematics, University of Leeds, Woodhouse Lane, LS2 9JT, Leeds, England; [email protected].

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5.5.1 Kronecker’s theorem under logical scrutiny . . . . . . . . . . . . . . . 5.5.2 The theorem of Eisenbud–Evans and Storch . . . . . . . . . . . . . . . Heitmann dimension: an exception that confirms the rule? . . . . . . . . . . .

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6 Appendix 6.1 Noetherian rings and excluded middle . . . . . . . . . . . . . . . . . . . . . . 6.2 Boundaries of basic opens for the Zariski spectrum . . . . . . . . . . . . . . . 6.3 Prime filters of rings and lattices . . . . . . . . . . . . . . . . . . . . . . . . .

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5.6

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Introduction

Hilbert’s programme is undoubtedly one of the main contributions of the last century to the foundations of mathematics. Ensuing G¨odel’s celebrated incompleteness results, Hilbert’s programme in its original formulation is nowadays widely deemed as unachievable.1 This notwithstanding, Hilbert’s work on foundations is still highly stimulating from a philosophical point of view; for example in recent years it has inspired instrumentalist proposals by Detlefsen and Field [38, 58]. The present contribution focuses on the impact some of Hilbert’s ideas have had on recent developments in mathematics. First of all it is worth recalling that the work of Hilbert and his school has been vital in shaping the landscape of mathematical logic as it is today; more specifically Hilbert’s programme has been instrumental to the birth of the central field of proof theory. Furthermore, Hilbert’s ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been source of motivation for the so–called reverse mathematics programme initiated by H. Friedman and G. S. Simpson (see section 2.3, [79, 53, 55, 123, 103, 6]). More recently Hilbert’s programme has inspired a new approach to constructive algebra in which strong emphasis is laid on the use of finite methods [10, 11, 12, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 37, 39, 40, 81, 83, 84, 85, 96, 97, 137]; see in particular [26, 30, 86]. This new trend in constructive algebra has especially focused on commutative algebra, and has sought an elimination of ideal objects in favour of more concrete, low–type ones. Such a shift has produced more elementary and more perspicuous proofs compared with the classical ones. Importantly, one of the motivations for the new approach is to ensure that proofs in commutative algebra have a clear computational significance. This new trend in commutative algebra constitutes the main topic of this note. Notwithstanding various differences among the modified Hilbert’s programmes that we briefly survey below, as well as dissimilarities between each of them and the original programme, we should like to emphasise the following common characteristics. First of all, they all constitute attempts by mathematicians to address foundational issues which arise straight from their own mathematical practice. As such, the mathematical component of these contributions appears to drive also the philosophical reflections, if not to overcast them. Secondly, all the programmes draw directly on Hilbert’s ideas, which constitute an explicit source of inspiration. Therefore, through these new programmes Hilbert’s legacy is very much alive today. 1

See Detlefsen [38] for a different perspective.

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2

Hilbert’s programme now and then

2.1

The original Hilbert’s programme

To set the stage for the subsequent discussion, we wish to recall some well–known themes related to Hilbert’s programme.2 Hilbert’s programme may be seen as having two fundamental components: the axiomatic method and finitary proof theory. Here a crucial role is played by the notion of consistency of a formal system.3 Simplifying considerably, Hilbert aimed at a formalisation of logic and the whole of mathematics, since in so doing we can express the entire ‘thought–content of the science of mathematics in a uniform manner’ and thus ‘the interconnections between the individual propositions and facts become clear’ [68, p. 475].4 The axiomatic method had the consequence of relieving our dependence from direct intuition: ‘a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument’ [68, p. 475]. In this context, therefore, the consistency of the axiom systems introduced gains fundamental importance. Hilbert thus aimed at showing, by exclusively finitistic means, that the formal systems encoding mathematics are consistent, that is to say, they are free of contradiction. The consistency of a body of mathematics did not need to be obtained directly, as it could instead be secured by first reducing the corresponding theory to another, more fundamental theory, and then proving the latter consistent. For example, in the case of geometry, Hilbert gave an arithmetical-analytical interpretation of its axioms, thereby reducing the question of its consistency to that of the consistency of the axioms for real numbers. The latter, however, needed to be obtained directly, as it was irreducible to a more fundamental problem. The consistency of the axioms for real numbers apparently was the ultimate problem for the foundations of mathematics according to Hilbert, so much so that already in 1900 it constituted his second problem in his famous address to the International Congress of Mathematicians. In fact, only with time and with the development of mathematical logic and proof theory by Hilbert and his school at the beginning of the 20th century, the problem of the consistency of the axioms for the real numbers became more precise.5 6 As an initial task, Hilbert’s collaborators attempted proofs of the consistency of arithmetic.7 2

The present note only highlights well-known aspects related to Hilbert’s programme, focusing on those which seem more relevant to the subsequent discussion. Regrettably, we here do no justice to the complexity of Hilbert’s thought (as well as of his collaborators’) nor take into due account the deep and rich literature on the subject. For an excellent recent survey on Hilbert’s programme, also in the light of the latest contributions in proof theory, we refer to Zach’s [139]; see also [88] and [120, 6], the latter especially with regard to the historical progression of the programme and the developments of metamathematics and proof theory. 3 We wish to highlight the purely syntactic character of the notion of consistency, and recall that, as clarified in [120, 6], Hilbert and Bernays had a fundamental role in bringing forth the nowadays familiar distinction between syntax and semantics. 4 Here and in the following we refer to the English translation of Hilbert’s work, as indicated in the references. 5 Niebergall and Schirn [92] suggest that Hilbert’s programme was formulated very imprecisely, so much so, that it can be seen as incapable of being refuted by G¨ odel’s incompleteness results. 6 It was observed by Hilbert and Bernays [69] that classical analysis can be formalised within second order arithmetic. Consequently, nowadays often proof–theorists identify analysis with second order arithmetic, and see the proof of consistency of the latter as the most challenging question stemming from Hilbert’s programme. This identification is however rather coarse, e. g. in view of the results highlighted in section 2.3. 7 It is well known that Ackerman proposed an erroneous proof of consistency of a version of analysis in his dissertation of 1924. He subsequently amended it, accepting suggestions from von Neumann. Ackermann and Bernays apparently believed the new version to constitute a proof of the consistency of number theory. Hilbert optimistically quoted these results in [68], suggesting that they could be easily extended to the whole of analysis. He wrote: “The method of W. Ackermann permits a further extension still. For the foundations of ordinary

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A proof of consistency of mathematics was deemed necessary by Hilbert also to fully justify the current mathematical practice, which had undergone substantial transformations during the nineteen century. In particular, one of the aims of the programme was to justify the use of the actual infinite in mathematics by a reduction of infinitary notions to finitary ones, as exemplarily expounded in [67]. The strategy was to obtain such a reduction by the axiomatic method and the newly devised mathematical enterprise of proof theory. Crucially, the proof of consistency of analysis had to be obtained by totally uncontroversial means, that is to say, according to Hilbert by strictly finitary means. We also wish to highlight that the axiomatic method and proof theory were considered as all encompassing tools which would systematically provide definitive answers to all foundational questions. More precisely, Hilbert hoped that his proof theory would enable him to: [. . . ] eliminate once and for all the questions regarding the foundations of mathematics, in the form in which they are now posed, by turning every mathematical proposition into a formula that can be concretely exhibited and strictly derived, thus recasting mathematical definitions and inferences in such a way that they are unshakable and yet provide an adequate picture of the whole science. ([68, p. 464]). Here the fundamental notion of a concrete object plays a crucial role, as clarified by the following quotation from [68]: No more than any other science can mathematics be founded by logic alone; rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. [. . . ] And in mathematics, in particular, what we consider is the concrete signs themselves, whose shape, according to the conception we have adopted is immediately clear and recognizable [68, p. 464–5].8 Finitary sequences of strokes and the concrete signs forming the formulas of a formal system are among Hilbert’s concrete objects, as they satisfy the requirement of complete surveyability and intuitive presentation. Alongside concrete objects Hilbert also envisaged finitary statements, the simplest examples of which are those expressing equality and inequality of numerals (see [68, p. 470], [67, p. 146]).9 In fact, elementary number theory held a special place in Hilbert’s proof theory10 , as its truths were considered provable through analysis his approach has been developed so far that only the task of carrying out a purely mathematical proof of finiteness remains. Already at this time I should like to assert what the final outcome will be: mathematics is a presuppositionsless science” [68, p. 479]. 8 Very similar remarks appear elsewhere in Hilbert’s writings, for example in [67, p. 142]. 9 Hilbert’s understanding of the numerals is prone to different interpretations. See e.g. [139] for a concise summary. 10 In [67] Hilbert calls number theory the ‘purest and simplest offspring of the Human mind’.

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contentual (inhaltlich) intuitive considerations and were thus prior to any form of logical inference. In fact, according to Hilbert, no contradiction can arise in contentual number theory as there is no logical structure in the propositions of this theory. Consequently, contentual number theory is secure and thus can constitute the basis for the justification of other parts of mathematics. It is interesting to note Hilbert’s attitude towards elementary number theory as well as the requirement of surveyability of the signs as witnessed by the quotation above, as they evoke the influence of Kronecker on Hilbert.11 We wish to recall that Kronecker’s ideas have also been a prominent source of inspiration for constructive mathematicians [17, 45, 83].12 For example, Bishop [17, p. 2] writes: The primary concern of mathematics is number, and this means the positive integers. We feel about number the way Kant felt about space. The positive integers and their arithmetic are presupposed by the very nature of our intelligence [. . . ]. The development of the theory of the positive integers from the primitive concept of the unit, the concept of adjoining a unit, and the process of mathematical induction carries complete conviction. In the words of Kronecker, the positive integers were created by God.13 Notwithstanding the crucial role of elementary number theory, ordinary mathematics is ripe of statements which go beyond contentual considerations. In fact, ‘even elementary mathematics goes beyond the standpoint of intuitive number theory’ [67, p. 145]. Therefore, Hilbert14 distinguished different kinds of formulas: first of all there are those to which there correspond contentual communications of finitary propositions (e.g. numerical equations or inequalities). These are unproblematic and the usual rules of classical logic safely apply to them. Then there are other finitary statements which however are problematic. For instance, in general the negation of a finitary statement need not be itself a finitary statement. Take for example the universal statement: for any a, a + 1 = 1 + a (where a is a meta–variable for a numeral). From a finitary perspective, this statement is ‘incapable of negation’ [67, p. 144] as ‘one cannot, after all, try out all numbers’ [68, p. 470]. For statements of this kind, the usual laws of classical logic do not hold. However, the usefulness of the Aristotelian laws of logic (in particular the excluded middle) is considered fundamental by Hilbert, and undisputable. According to Hilbert, relinquishing classical logic would force us to abandon the ‘tremendous progress’ made in mathematics so far. Now, the difficulty which arises here can be overcome by the introduction of a new kind of propositions, called ideal propositions. These are formulas which in themselves mean nothing but which represent the ideal objects of the theory and are introduced ‘in order that the ordinary laws of logic would hold universally’ [67, p. 146]. In [67, 68] Hilbert used the example of algebra and the use of variables in algebra 11

It is well known that beyond the quite different perspectives on the foundations of mathematics of the two influential mathematicians, there is a component of Kronecker’s legacy in Hilbert’s thought. This is for example elicited by Weyl in [135]. 12 We could also mention at this point that further important source of philosophical inspiration for many constructive mathematicians are Kant, Poincar´e and Husserl (see for instance [17, 89, 83, 131]). 13 We wish to recall that Bishop’s text proceeds as follows: ‘Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.’ 14 Here we closely follow [67].

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to clarify this point. For example, instead of the concrete formula a + b = b + a, where a and b stand for particular numerical symbols, it is practice in algebra to prefer the formula: a + b = b + a. The latter, however, has no meaning of its own, as it does not express a finitary statement, it is not an immediate communication of something signified. On the contrary, it is a formal structure from which we can obtain the corresponding finitary statements by substitution of the variables by appropriate numerical symbols. This is to say, one can derive from ideal formulas other ones which do have meaning. Hilbert clarified that there is one indispensable condition attached to the use of the method of ideal elements: ‘extension by the addition of ideal elements is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain.’ [68, p. 471] We wish once more to stress the role of the formalisation of mathematics. Hilbert thought that by formalising mathematics we could replace abstract concepts and complex derivations by contentual investigations, that is by logic–free manipulation of formulas according to specific and circumscribed rules. In [68, p. 471] Hilbert wrote: ‘But a formalised proof, like a numeral, is a concrete and surveyable object. It can be communicated from beginning to end.’ Hilbert claimed, quite optimistically, that once the formalisation of mathematics had been carried out, then a proof of consistency would have easily followed. For example in [68] he writes that consistency statements amount to showing that a certain formula (for example 0 6= 0) can not be derived by our axioms and rules. However, that the end formula of a derivation ‘has the required structure, namely 0 6= 0, is also a property of the proof that can be concretely ascertained’ [68, p. 471]. Hilbert also once more stated that a finitary proof of consistency ‘can in fact be given, and this provides us with a justification for the introduction of our ideal propositions.’ [68, p. 471]. In fact, a proof of consistency would have enabled to fully justify the use of classical reasoning applied to them. The question of what counted as finitary means of proof for Hilbert and his close collaborators has given rise to different interpretations. Hilbert and Bernays give examples of finitary operations which can be characterised in formal terms by the concept of primitive recursive functions. But as emphasised for example in [127, 138] the Hilbert school did use and accept methods which went beyond primitive recursion. Nevertheless, it is not clear whether they were aware of that and if they would have still accepted them if they were. Tait [126] has convincingly argued that finitism can be formally captured by a fragment of Peano Arithmetic (PA), which is known as Primitive Recursive Arithmetic (PRA). This is obtained by restricting PA’s induction principle to quantifier–free induction.15

2.2

Generalised and relativised Hilbert’s programmes

We recall that Hilbert’s programme is often understood as grounded on the following requirements16 : • At least all of PA is to be proved consistent; in fact, the aim is a proof of consistency of all of mathematics. 15

A quite restrictive view of finitism has been proposed by Parsons [95], for which finitism is better captured by taking only addition and multiplication as starting functions and additionally allowing for bounded induction only. A more permissive characterisation of finitism has been proposed by Kreisel [79], according to which finitary reasoning is captured by a system having the same strength as PA. 16 See [38] for a divergent view. See also [92].

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• The proof of consistency ought to be given by exclusively finitary methods. If Hilbert’s programme is so understood, G¨odel’s second incompleteness theorem constitutes a fundamental obstacle for the programme, provided that all finitary arguments can be formalised within PA. At first, G¨odel and Bernays contemplated the idea that it might still be possible to account for the consistency of PA by employing methods which were not formalizable in PA but which could still be considered finitary.17 Nevertheless, already from the mid 1930s it was widely accepted that all finitary reasoning could be formalised in PA. In fact, as we have seen above, it is nowadays widely acknowledged, following Tait [126], that PRA suffices. Soon after G¨ odel’s incompleteness results, it was realised (e.g. by Bernays) that an enlarging of the methods of proof theory would have allowed the survival of (a modified) Hilbert’s programme. In fact, it was suggested that ‘instead of a reduction to finitist methods of reasoning, it was required only that the arguments be of a constructive character, allowing us to deal with more general forms of inferences.’ [14, p. 502]. Bernays remarks indicate the possibility of a modification of Hilbert’s programme obtained by relaxing the methods used in the consistency proofs, which can now be taken to be broadly constructive rather than finitary. Following e.g. [139] we call this kind of modified Hilbert’s programme ‘generalised Hilbert’s programme’.18 Another kind of modification of Hilbert’s programme, proposed by Kreisel and then especially brought forward by Feferman, is usually referred to as ‘relativised Hilbert’s programme’ [53]. Contrary to generalised Hilbert’s programmes, relativised Hilbert’s programmes may be seen as pursuing local rather than global projects, as clarified below. As to generalisations, these are exemplified by Gentzen’s consistency proof for arithmetic.19 Here crucial role is played by the use of Transfinite Induction (TI) up to the (countable) ordinal 0 .20 In modern terms, Gentzen’s result can be rephrased as a proof of the consistency of PA given on the basis of the system PRA plus TI up to 0 . In other terms, according to Tait’s identification of PRA as coextensive with finitary reasoning, the only assumption in the proof of consistency of PA which goes beyond finitism is that of TI up to 0 . What is significant here is that TI needs to be applied only to predicates which can be finitistically described (i.e. they are primitive recursive).21 Crucially, the reference to ordinals can be recast within arithmetic by using so called ordinal notation systems. These give representations of countable ordinals by means of natural numbers. For example, Cantor’s normal form theorem provides a natural ordinal representation system which can be used in the proof of consistency of arithmetic.22 17 As quoted in [54], G¨ odel wrote in [65, p. 138–9, 195]: ‘I wish to note expressly that [this theorem does] not contradict Hilbert’s formalistic viewpoint ... it is conceivable that there exist finitary proofs that cannot be expressed in the formalism of P ...’. Here P refers to G¨ odel’s finite theory of types of [64]. 18 This is also called ‘extended Hilbert’s programme’ in [56]. 19 We here loosely refer to ‘Gentzen’s proof’, even if, as it is well known, Gentzen gave a number of distinct proofs of the consistency of arithmetic, as highlighted for example in [134]. See also [21] for a formal comparison between Gentzen’s 1938 proof and the rendering of his result by means of infinitary systems (` a la Sch¨ utte). ω 20 The ordinal 0 is defined as sup{ω, ω ω , ω ω , . . . } = least α (ω α = α). 21 In fact, they are elementary computable [105]. 22 Cantor normal form theorem states the following: for every ordinal α > 0 there exist unique ordinals α0 > α1 > . . . > αn such that α = ω α0 + · · · + ω αn .

As a consequence of the theorem, ordinals α < 0 have normal form with αi < α for each 0 6 i 6 n. Also, each exponent has Cantor normal form with yet smaller exponents. As this process must terminate, ordinals < 0 can be encoded by natural numbers. We observe, furthermore, that for systems proof–theoretically stronger

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Considerations of this kind have given rise to a reading of the proof of consistency in terms of an extended form of finitism, for example by Gentzen [61] and Takeuti [128]. We wish to expand slightly on Takeuti’s views as presented in [128, §11], as, although perhaps philosophically little convincing, they raise some interesting points. There the author proposes an extended form of finitism (called Hilbert–Gentzen finitism), and distinguishes this view from the one which resorts to constructive methods in consistency proofs. Takeuti in fact recalls that intuitionism makes some abstract assumptions, an example of which is given by the notion of construction or proof which figures in the Brouwer–Heyting–Kolmogorov explanation of the connectives and quantifiers. Abstract assumptions, Takeuti suggests, are to be avoided as much as possible. However, one can justify a form of extended finitism, which only makes abstract assumptions of a restricted, more concrete kind. These assumptions are called Gedankenexperimente (thought experiments) by Takeuti. In Gentzen’s proof, these are necessary to convince oneself that a certain ordering is in fact a well–ordering. Quoting from [128, p. 96]: A Gentzen–style consistency proof is carried out as follows: 1. Construct a suitable standard ordering, in the strictly finitist standpoint. 2. Convince oneself, in the Hilbert–Gentzen standpoint, that it is indeed a well–ordering. 3. Otherwise use only strictly finitist means in the consistency proof. Gedankenexperimente are crucially needed in performing step (2). According to Takeuti, the distinguishing feature of these Gedankenexperimente is that they only act on concretely given sequences. Hence, according to Takeuti, they are not absolutely abstract as some of the intuitionist’s assumptions. In other terms, Takeuti’s extended finitism can be characterised as Hilbert’s finitism plus Gedankenexperimente, which are believed to represent a minimal form of ideal component. Although quite suggestive, Takeuti’s notion of Gedankenexperimente is probably not sufficiently clear, nor it is clear how to distinguish between absolutely abstract assumptions and more concrete ones [52]. However, his discussion well highlights a typical attitude of many proof–theorists, who are always keen on clarifying the ingredients of a proof, and so pinning down precisely where specific, stronger assumptions pop up. As we shall see below, this is a feature traditionally associated to work in the area of proof–theory, which is making its way within other areas of mathematics, for example through the work in reverse mathematics and, now, constructive algebra. In fact, Takeuti’s discussion highlights further general difficulties facing the proof–theorist’s attempts of framing his work on Hilbertian grounds. With Gentzen’s work, that branch of proof theory known as ordinal analysis came into existence. A central theme of ordinal analysis is the classification of theories by means of ordinals. This is achieved by the assignment of proof–theoretic ordinals to theories, measuring their consistency strength and computational power.23 In simple terms, such an ordinal analysis attaches ordinals in a given representation system to formal theories. Beyond its impressive technical achievements, difficulties than PA, one needs more powerful ordinal representation systems. In fact, fundamental and challenging work has been necessary to represent the much higher ordinals necessary for the analysis of more complex fragments of analysis. See e.g. [105] for a gentle tour of ordinal analysis. 23 For example, Gentzen’s consistency proof allows us to assign the ordinal 0 as proof–theoretic ordinal to PA. This is on the basis that PA proves TI for each ordinal strictly less than 0 but it does not prove TI for 0 .

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arise here with respect to the philosophical underpinnings of ordinal analysis. This branch of proof theory, in fact, is often presented as a way of justifying larger and larger fragments of second order arithmetic (i.e. analysis) in constructive terms. However, first of all the very notion of constructivity is often not sufficiently spelled out (similarly to Takeuti’s notion of Gedankenexperiment), and one may claim therefore that this reduction is not totally justified.24 Furthermore, over the years, the work in proof theory has become more and more complex, as witnessed for example by recent results on the consistency of strong subsystems of second order arithmetic [101, 104, 2, 3]. Consequently, it becomes more and more challenging for a non–expert to grasp what in these proofs exceeds Hilbert’s finitism, and thus to assess their contribution to a reductionist programme of this kind [54]. For example, Feferman [54] recalls that The crucial question is: In what sense is the assumption of TI(α) justified constructively for the very large ordinals α used in these consistency proofs? Indeed, on the face of it, the explanation of which ordinals α are used appeals to the very concepts and results of infinitary set theory that one is trying to account for on constructive grounds. Kreisel and Feferman have proposed an alternative route based on Hilbertian themes: relativised Hilbert’s programmes. In the case of relativised Hilbert’s programmes, the allencompassing task of providing a finitist consistency proof is replaced by the aim of explaining parts of ideal mathematics in terms of more justified ways of reasoning in some sense. In practice this often amounts to reducing one formal system to another more elementary one, as clarified below. In addition, Kreisel raised objections to the too strong emphasis on consistency proofs as the main tool and focus of generalised Hilbert’s programmes. In relativised Hilbert’s programmes, consistency proofs are still a tool at hand, although often in the form of relative consistency; however, other proof–theoretical methods (e.g. functional interpretations) are envisaged. In fact, the principal aim is to use proof theory to make explicit the additional knowledge provided by those proofs. A further idea is that proof theory clarifies ‘what rests on what’ in mathematics [56]. Here the notion of proof–theoretic reduction plays a fundamental role. This can briefly be described as follows. One works on the basis of a reference system, say U ; in practice U can be taken to be PRA. Given two theories S and T which contain U , let Φ be a primitive recursive class of formulas common to the languages of both theories. In addition, Φ should contain the closed equations of the language of U . We say that S is proof–theoretically Φ– reducible to T if in U one proves that given any formula ϕ belonging to the class Φ, every proof of it in S can be transformed via a primitive recursive method into a proof of the same in T . More precisely, S is proof–theoretically Φ–reducible to T , if there exists a primitive recursive function f such that U ` ∀ϕ ∈ Φ∀x (Proof S (x, ϕ) → Proof T (f (x), ϕ)), where Proof V (y, z) expresses that y codes a proof in the theory V of the formula coded by z. Note that if S is proof–theoretically Φ–reducible to T , then S is conservative over T for Φ, in the sense that: if ϕ ∈ Φ and S ` ϕ then T ` ϕ. 24

See however section 2.3.

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If false statements such as 0 = 1 belong to Φ, then S is relatively consistent to T . With the notion of proof–theoretic reduction at hand, one can see in exact terms how some forms of abstract mathematics can be reduced to more elementary ones, by looking at formal systems which codify those forms of mathematics and studying their relationship with more elementary ones. Following [53, p. 364], the pattern can be summarised as follows: A part of mathematics M is represented in a formal system T1 which is justified by a foundational or conceptual framework F1 . T1 is reduced proof–theoretically to a system T2 which is justified by another, more elementary such framework F2 . For example, Kreisel and Feferman considered not only the reduction of the infinitary to the finitary but also of the nonconstructive to the constructive and of the impredicative to the predicative. In fact, a substantial amount of work has so far been accomplished allowing for the reduction of theories which codify a considerable amount of ideal mathematics to more elementary ones, as for example clarified in [53, 54, 56].

2.3

Contemporary perspectives of Hilbert’s programme

A related line of defense of roughly Hilbertian themes has appeared in writings of Simpson, pertaining to Friedman and Simpson’s Reverse Mathematics programme [122, 123]. Simpson claims here to have obtained a partial realisation of Hilbert’s programme, since ‘one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best–known nonconstructive theorems.’ [122, p. 349] The purpose of Reverse Mathematics is to discover which set existence axioms are needed in order to prove specific theorems of ordinary or core mathematics. Often the theorems turn out to be equivalent to the axioms; hence the slogan ‘reverse mathematics”. This programme uses the language of second order arithmetic and it has isolated five main subsystems of it that frequently occur as the reversals of mathematical theorems. To classify a mathematical theorem one usually shows that it is equivalent, on the basis of the next weaker system, to the principal set existence axiom of one of these five systems. In this way one shows which set existence axioms are actually needed for a specific mathematical construction. As a by–product of the analysis, one often obtains more complex but more informative proofs, compared with the standard ones. Surprisingly, results from reverse mathematics have revealed that quite a large part of infinitary mathematics can be reduced to finitistic reasoning. Of fundamental importance in this context is a subsystem of second order arithmetic introduced by H. Friedman and known as WKL0 (the initials referring to Weak K¨onig’s Lemma). Although mathematically rather strong, this system is proof–theoretically very weak; in fact it is conservative over PRA with respect to Π02 sentences. This fact prompts Simpson to claim that ‘any mathematical theorem which can be proved in WKL0 is finitistically reducible in the sense of Hilbert’s programme’ [122, p. 354]. In other terms, many theorems which can be proved by use of infinitary techniques and concepts can also be proved elementarily, with the resulting elimination of the relevant infinitary components.25 We also wish to recall here that recently Ishihara, Veldman and others have initiated a programme which goes under the name of constructive reverse mathematics (see for example 25

We observe that reverse mathematics goes well beyond the aims of a partial realisation of Hilbert’s programme as it also studies mathematical equivalents of subsystems of second order arithmetic which are much stronger than WKL0 .

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[72, 73]). It takes inspiration from Friedman and Simpson’s programme, but differs from it for its scope of action. It builds on the known fact that Bishop-style constructive mathematics is compatible with intuitionistic, recursive and classical mathematics [20]. In fact, each one of these kinds of mathematics may be framed as Bishop-style mathematics plus some specific principles [20]. The idea then is that the privileged standpoint of the constructive mathematician allows him to compare notions and results which are possibly incompatible with other mathematical traditions (e. g. classical or intuitionistic). Thus the constructive programme aims at classifying not only theorems in classical mathematics but also theorems in constructive, recursive constructive, intuitionistic mathematics. Constructive reverse mathematics appears thus as a very promising path in the direction of an understanding of the mathematical practice in constructive terms. It also highlights a new trend within constructive mathematics, by introducing a very fine attention to the ingredients of a proof. In this respect constructive reverse mathematics bears some similarities with the algebraists programme to be highlighted below. The results from reverse mathematics might appear quite surprising to the general mathematician, who might wonder if a careful formalisation as the one needed to obtain these results is worth the effort. In fact, as already indicated above and also highlighted for example in [5] and [6], starting from Gentzen’s pioneering work, proof theorists have developed a tradition of trying to keep to a minimum all assumptions they make, and they have also studied weak as well as relatively strong theories.26 In [5] the focus of attention is a system which is even weaker than PRA. This is called Elementary Arithmetic (EA) and is so weak, for example, that it does not prove the totality of an iterated exponential function; in fact, its consistency can be proved in PRA, thus, according to Tait, by finitistic means. Again, as already clarified by Feferman and others, among proof–theorists ‘the general feeling is that most ‘ordinary’ mathematics can be carried out, formally, without using the full strength of ZFC’ [5, p. 258]. Avigad observes that although so weak, from the point of view of finitary number theory and combinatorics EA turns out to be surprisingly robust. So much so that Harvey Friedman has made the following Grand Conjecture:27 Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in elementary arithmetic. In [5] the author presents some concrete examples of significant theorems (Dirichlet’s theorem on primes in an arithmetic progression and the prime number theorem) which although prima facie requiring a good amount of abstract notions, can be shown to hold in systems reducible to EA. Although Avigad recognises that a solution (in one direction or the other, or possibly a mixed one) to Friedman’s conjecture is far from been reachable today, he also suggests some further reasons for working in weak theories.28 The act of ‘mathematizing with one’s hands tied’ can often yield new proofs and a better understanding of old results. It can, moreover, lead to interesting 26 Note that, even the strongest subsystems of second order arithmetic studied for example in [101, 104, 2, 3] are rather weak if compared with the full power for example of ZFC. 27 The conjecture was posted to the Foundations of Mathematics discussion group [FOM] on April 16, 1999. 28 Similar arguments have been put forward by Kreisel and Feferman. See for example [56].

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questions and fundamentally new results, such as algorithms or explicit bounds that are absent from non-constructive proofs. [5, p. 272] One could say that a verification of Friedman’s conjecture would show that there is a precise sense in which the infinitary methods found in the Annals of Mathematics can ultimately be justified relative to finitary ones. In fact, elementary arithmetic appears to represent a notion of finitism which is quite uncontroversial; indeed, it can account for the methods of proof which would have been accepted by Kronecker.29 30 We ought also to mention at this point the line of research often called proof mining, carried out by Kohlenbach and others aiming at obtaining explicit bounds or rates of convergence from classical proofs especially in analysis [78]. We conclude this sketchy panoramic on the influence of Hilbert’s programme on contemporary proof theory with a brief hint at some ideas presented by Rathjen in [103]. Taking inspiration from both generalised and relativised Hilbert’s programmes, Rathjen [103] proposes a so–called Constructive Hilbert’s programme, aiming at a constructive justification of a substantial portion of classical mathematics. A crucial aspect of Rathjen’s approach is that he clearly specifies what ‘constructive’ means, by referring to a specific framework of theories: Martin–L¨of type theory (MLTT). This is usually considered the most satisfactory foundation for constructive mathematics. Rathjen argues that due to the above mentioned work of Feferman as well as the results in reverse mathematics, one can single out a suitable subsystem of classical second order arithmetic as formalising ordinary classical mathematics. Let’s call this subsystem T . The claim is that large parts of infinitistic mathematics can be developed in T . The author comes to the conclusion that ordinary mathematics is demonstrably consistent relative to MLTT, since a strong version of this latter theory proves the consistency of T . As the consistency proof for T is couched in terms of MLTT, one thus makes use of constructive and predicative rather than finitistic methods. Rathjen’s article touches some very intriguing issues relating to the foundations of constructive mathematics and in particular constructive type theory. Significant extensions of Martin–L¨of type theory have been proposed in recent years by Palmgren, Rathjen and Setzer [93, 106, 102, 119]. Questions then arise on how to justify these powerful extensions and how far can one carry on this kind of extension process while remaining within a given constructive framework of ideas. Rathjen’s paper proposes an analysis of MLTT and its limits from an external, classical point of view, as ‘a demarcation of the latter is important in determining the ultimate boundaries of a constructive Hilbert program. The aim is to single out a fragment of second order arithmetic or classical set theory which encompasses all possible formalisations of Martin-L¨ of type theory.’ Rathjen presents an engaging conjecture regarding MLTT’s limits. We believe that more work still needs to be carried out to fully understand these extensions of type theory also from an internal, fully constructive point of view, and thus to assess their role for the reductionist programmes of the proof–theorists [56]. 29

Avigad’s paper [5] goes beyond the scope of this survey, by also addressing some fundamental philosophical questions which arise in connection with the proof–theoretic attitude to mathematics. 30 Here we wish to stress the deep similarities of attitude with the new programme in constructive algebra to be surveyed in the next section.

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3

Finite methods for constructive algebra

In this note we are particularly interested in the influence Hilbert’s programme is having for a new course in constructive algebra, especially through the work of Coquand, Lombardi and others [10, 11, 12, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 39, 40, 81, 83, 84, 85, 96, 97, 137]. The first observation we wish to make is that the authors often frame their work explicitly as a contribution to Hilbert’s programme. In fact, in [26] Coquand writes that recent work in commutative algebra “can be seen as achieving a partial realisation of Hilbert’s programme” in such a field. The abstract of Coquand and Lombardi’s [30] reads accordingly: “Recent work in constructive mathematics shows that Hilbert’s programme works for a large part of abstract algebra”. Similar claims are to be found in a number of their articles and presentations. Before clarifying the aims and methods of this undertaking, we wish to recall that constructive algebra is part of the wider field of (Bishop’s style) constructive mathematics. Though its origins can be traced back to Brouwer’s pioneering ideas and Heyting’s realisation of the importance of intuitionistic logic, constructive mathematics had its rebirth starting from Bishop’s publication of Foundations of constructive analysis [17] (see also [18, 19]). Bishop’s style constructive mathematics has often been characterised as mathematics based on intuitionistic logic [20, 109, 19]. Many constructive mathematicians further perceive their discipline as incorporating a form of predicativity, which is usually referred to as generalised predicativity. This is a quite generous notion of predicativity [57]. More specifically, for example, the unrestricted use of powerset is considered as unjustified according to this concept of constructivity; however, inductive definitions are allowed, also in their generalised form. Furthermore, we wish to recall that, following Richman’s recent appeal for a mathematics without choice [110, 111], lately more efforts have been dedicated to developing a choice–free constructive mathematics. As to the reasons for pursuing constructive mathematics, we could recall that already Bishop saw this as a necessity if one aims at recovering the computational content of mathematics that may be lost through the use of classical logic (see the introduction to [17] and also [19]). In fact, the computational significance of constructive mathematics is one of the grounds for the more widespread recognition this discipline is receiving nowadays. Certainly, as is well–known, constructive proofs have a direct computational content and thus are amenable to program extraction; and, crucially, programs extracted from constructive proofs are automatically correct. The work we are interested in is especially focused on commutative algebra. As a partial revival of Hilbert’s programme it can be seen as having two fundamental components: the elimination of ideal objects, and the use of finite methods. The strategy is to analyse classical theorems in commutative algebra and show that when one proves a concrete statement, one can often eliminate the use of ideal objects, and obtain a purely elementary proof. The elimination of the ideal objects is frequently accomplished by finite approximations of infinite objects, and by appealing to a more syntactical and low–type description of the classical notions. Indeed, the aim is often to prove low–type or concrete statements without resorting to notions which are formulated in a higher type level than that of the statement itself. One motivation is that constructive proofs carried out on low type levels are particularly suited for a formalisation within computer-assisted proof systems. It might be interesting to ponder a little on the choice of the field of commutative algebra. Commutative algebra, in fact, is a pivotal corner of contemporary mathematics, which by 13

and large—among other things—has set the grounds for the eventual settling of the famous conjecture known as Fermat’s Last Theorem. But what does the issue of type levels mean in this context? Cannot many theorems of algebra be expressed by something as simple as a finite number of polynomial equations with rational coefficients and in only finitely many indeterminates, whereas to speak for example of analysis requires from the outset a permanent talk of genuinely infinite objects such as real numbers, continuous functions, and the like? Isn’t therefore algebra a priori much easier to deal with than analysis, from the angle of a (modified) Hilbert’s programme? The answer to all these questions would have been in the affirmative before Dedekind’s times, and still for Kronecker’s work. However, abstract concepts and methods characteristic of modern algebra were put forward by Dedekind; profited from the rise of set theory and topology; and became most powerful in the hands of Hilbert and his followers (E. Noether, Krull, and others). It is this abstract character which causes problems if revisited from the perspective of a modified Hilbert’s programme; of course not all theorems are affected, but quite a few of the short and elegant proofs done by use of ideal objects. The prime example of an ideal object in commutative algebra is presumably the concept of an ideal that Dedekind has made of Kummer’s “ideal numbers” [41, 42, 43, 44]. Section 5.3 will explain more in detail the difficulties which arise with this notion, and the solution which has been proposed in order to deal with it in a more concrete way. Anticipating a little the contents of that section, we can recall for example that classically the notion of prime ideal of a ring R is usually rendered by appeal to the notion of a subset of R, and thus requires to step into a higher type level compared with the level where the elements of the ring reside. In addition, often proofs of existence of prime ideals involve applications of Zorn’s lemma, which is classically equivalent to the axiom of choice. More specifically, the concept of prime ideal is essential for the fundamental concept of the Zariski spectrum Spec(R) of a commutative ring R.31 In constructive algebra, following Joyal [77], Spec(R) is represented as a distributive lattice, L(R), the lattice generated by the standard basis of the Zariski topology (section 5.3 will clarify why this is in fact possible). The pleasing result of such a device is that all reference to subsets of R (especially to prime ideals) is thus avoided, and in fact this L (R) is situated at the very same type level as R is. With clear reference to Hilbert, the constructive algebraist stresses the direct effective description of the lattice L (R) (rather than defining it as the lattice of compact opens of Spec(R)) and the related fact that we can work with it by simple manipulation of purely symbolic expressions. We have seen above that Kreisel and Feferman’s relativised Hilbert’s programme aims at explaining what are perceived as more complex forms of mathematics in terms of more elementary ones in some sense. For example, they show what is required to explain the infinitary (nonconstructive, impredicative) in terms of the finitary (constructive, predicative). Here, too, the new programme can also be seen as an attempt to show how to give a constructive explanation of some of the ideal elements which are found in commutative algebra. Thus, for example, the lattice L(R) constitutes a way of understanding constructively the spectrum Spec(R) of the ring R. It is worth recalling that in this enterprise the constructive rendering of a classical notion or a constructive proof are not totally apart from the original classical objects. Already Bishop highlighted the heuristic role of classical proofs for constructive mathematics [17, p. 31

The points this topological space is made of are indeed nothing but the prime ideals of R.

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x]: Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof. In the case of the work by Coquand, Lombardi et al here under consideration, the authors also highlight the fact that often the classical proof guides them in finding the constructive argument, as one can in many cases use the ideas contained in the classical argument in an essential way.32 As for Bishop, here, too, reformulating the statement in an appropriate way constitutes a very important part of the whole constructive enterprise. Once the goal statement has been expressed in concrete terms, then one knows that there must be an elementary proof of it. Importantly, the resulting arguments are in general better proofs inasmuch as they not only are more elementary, and thus more transparent, but also have a clear computational content.33 For Hilbert and his school, consistency was a fundamental instrument in justifying idealistic mathematics. As we have already seen, the prominent role of consistency proofs for Hilbert’s programme had been criticised by Kreisel and Feferman, and consequently had a more limited role within their relativised programmes. It is interesting to remark that here, too, there is not such a general and all–accomplishing tool as a consistency proof. On the contrary, the work carried out so far has looked at some relevant specific examples, in a case– by–case manner. Proof–theoretic techniques, as for example double negation interpretation or cut–elimination can be used for parts of the proofs, but in general each problem requires a new approach and new ideas. Two classical results can be used as heuristic tools for obtaining a constructive proof from a classical one: the completeness theorem for first-order logic and Barr’s theorem on geometric logic. By completeness we know that if a first-order formula has a proof, possibly containing ideal components, then we can also give a purely first-order proof of it. In fact, if a statement is furthermore formulated equationally, Birkhoff’s completeness theorem for equational theories [16] ensures that there is a purely equational proof of it. Barr’s theorem refers instead to a particular kind of formula which is known as geometric formula (see section 4 below for details). More precisely, this theorem says that if a geometric sentence is derivable from a geometric theory in classical logic, possibly also by application of the axiom of choice, then it is also deducible from the same theory by intuitionistic logic and without appeal to the axiom of choice. We need to stress again that the full forms of both completeness and Barr’s theorem have classical proofs; consequently they cannot be used in general as automatic translations from classical to intuitionistic theories. In fact, they rather are heuristic tools: to know that an elementary proof ‘must exist’ helps to find it. At this point it is in order to underline the importance of an analysis of the logical complexity of statements. As already indicated, much effort in this modified Hilbert’s programme is dedicated to analysing the logical complexity of given classical theorems and attempting to reformulate them in such a way that their complexity is reduced as much as possible. The more general perspective behind this kind of analysis can be summarised as follows: 32 This is why with [28, 84] a series of papers was begun whose titles contain the phrase “Hidden constructions in algebra” or “Constructions cach´ees en alg`ebre abstraite”. 33 This bears similarities with the arguments e.g. given in [5], where the author highlights the benefits of “mathematizing with one’s hands tied”.

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We see this as a modest, but significant, first step towards the general program of analyzing logically contemporary algebraic geometry, and classifying its results and proofs by their logical complexity.[26]

4

Geometric formulas and dynamical proofs

As already observed in the mid 1970s [136, 87], first–order and geometric formulas play a prominent role in constructive algebra, and so do the so-called coherent formulas which are simultaneously geometric and first-order. The limited logical complexity of geometric formulas has made it possible to single out a peculiar notion of proof, called dynamical proof [15, 37]. A dynamical proof is represented by a tree, and is seen as “logic–free” and elementary. In the case of coherent theories the proofs are even represented by finitely branching trees. We now sketch the road from geometric formulas to dynamical proofs, closely following [24, 30]. In algebra atomic formulas are usually (in)equalities between terms. For example, in the language of rings they are of the form s = t, while in the case of lattices it is convenient to admit also atomic formulas of the form s 6 t. A positive formula is built from atomic formulas by means of _ >, ⊥, ∧, ∨, ∃, , W where infinite disjunction is often restricted to countable index sets. Note that neither implication → nor the universal quantifier ∀ may be used to build up positive formulas, let V alone infinite conjunctions . A geometric formula is of the form ∀x(ϕ → ψ) with positive subformulas ϕ and ψ. Note that the universal quantifier ∀ is only allowed in the outmost position, e.g. expressing universal closure, while → is only permitted in the next position. For example, if ϕ is positive, then ϕ and ¬ϕ are geometric: the first is equivalent to > → ϕ, and the second is defined as ϕ → ⊥. Crucially, neither ϕ ∨ ¬ϕ nor ¬¬ϕ → ϕ are geometric, for ¬ϕ is not positive: it contains an implication. A geometric theory is a theory whose axioms are all geometric sentences. A formula is W V first-order if it has no occurrence of infinite or ; and a coherent formula is a first-order geometric formula. A Horn formula is of the form θ1 ∧ . . . ∧ θn → θn+1 where all the θi are atomic formulas; whence every Horn formula is coherent. Equational theories consist of Horn sentences. Examples of equational theories are the theory of (commutative) rings34 and the theory of (distributive) lattices. The notion of an integral domain however cannot be captured by an equational theory for one needs to have the coherent axiom xy = 0 → x = 0 ∨ y = 0 . Although the crucial axiom of a field fails to be a geometric formula if it is put as ¬x = 0 → ∃y.xy = 1 , it becomes even coherent once it is rephrased in the classically equivalent way x = 0 ∨ ∃y.xy = 1 . 34

See the beginning of section 5.1 for a quick reminder of the relevant notions.

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(1)

Here are two more examples of definitions that can be done by geometric formulas. First, to express that a ring element a is nilpotent one needs a positive formula that is not coherent: _ an = 0 . n∈N

But that the ring A is reduced (i.e., 0 is the only nilpotent element) can even be put as a Horn formula: x2 = 0 → x = 0 . Wraith [136] underlines that geometric formulas are relevant because of Barr’s theorem, which was formulated and proved within topos theory [13]: Theorem 1 (Barr) For every geometric theory T and geometric sentence θ, T + AC `c θ ⇒ T `i θ . In other words, if T proves θ with classical logic and the axiom of choice, then T proves θ with intuitionistic logic and without choice. A coherent formula can further be characterised as (equivalent to) a finite conjunction of formulas of the form → ϕ ∨ . . . ∨ ∃− ϕ0 → ∃− x x→ (2) 1 1 n ϕn where every ϕi is a finite conjunction of atomic formulas without parameters. This normal form of coherent formulas is related to the notion of a dynamical proof [15, 24, 37] as follows. Atomic sentences are called facts. Let T be a coherent theory, and θ ≡ θ1 ∧ . . . ∧ θn → θn+1 a Horn sentence with facts θi . To deduce θ from T a finitely branching, finite proof tree can be generated where • a node is a set of facts, together with their parameters, known at a time; • the root consists of the given facts θ1 , . . . , θn ; • a leaf either contains θn+1 , or else is the empty set ∅ and thus represents ⊥. The facts belonging to a node represent the state of knowledge at a certain time, and knowledge is accumulated with time unless one arrives at ∅. The immediate successors are generated according to the inference rules that correspond to the axioms of T written in the normal form (2). Here disjunction is understood as case distinction, with every case giving rise to an immediate successor, and executing the existential quantifier means adding a new parameter; whence in particular all the inference rules are intuitionistically valid. Consider, for instance, a geometric theory T that contains a relativised field axiom such as θ0 → x = 0 ∨ ∃y.xy = 1 . (3) Now if N is a node to which the fact θ0 and the parameter a belong, then N can be endowed with two immediate successors: one of them is N endowed with the fact a = 0; the other one is N enriched with a fresh parameter b and the fact ab = 1. 17

Interestingly, a computational interpretation of the notion of dynamical proof is possible. Here geometric axioms can be seen as subprograms, and branches in a proof tree as runs of a program. The former can be explained along the lines of the foregoing example: the relativised field axiom (3) corresponds to a routine, only applicable if the fact θ0 holds, which for any given a tests whether a = 0 and, if the result is negative, produces b such that ab = 1. Apart from this interpretation, there are strong metamathematical reasons for considering dynamical proofs [30]:35 The crucial point is that this notion of dynamical proof is complete for deducibility in a coherent theory [15, 24, 37], and that a dynamical proof uses only intuitionistically valid inference steps. Barr’s theorem [for coherent theories and sentences] [. . . ] is a simple consequence: if a coherent sentence is deducible from a coherent theory in classical logic, even with the axiom of choice, it is a semantical consequence of the theory, and so, by completness, it can be derived by a dynamic proof, which is intuitionistically valid. One can speak of a dynamical proof also in the more W general case of a geometric theory T that indeed contains countably infinite disjunctions , in which case the proof tree is infinitely branching, and completeness as above is still valid. A connection to Hilbert’s programme is, with facts as concrete statements, as follows [30]: A dynamic proof can be seen as a “logic-free” and elementary way to derive new concrete statements from [. . . ] a given collection of concrete statements. By completeness, we know that if we can derive a concrete statement from this theory with the use of ideal methods (typically using Zorn’s lemma), there is also an elementary derivation. [. . . ] From a constructive perspective, however, this requires a warning to be sounded [30]: Both the completeness theorem and Barr’s theorem are purely heuristic results from a constructive point of view however. Indeed, they are both proved using non constructive means, and do not give algorithms to transform a non effective proof to an effective one. But this notwithstanding they seem to work quite well as heuristics [30]: In practice however, in all examples analysed so far, it has been possible to extract effective arguments from the ideas present in the non effective proofs. We think that our work, complementary to the work done in constructive mathematics [45, 90] or in Computable Algebra [124], provides a partial realisation of Hilbert‘s program in abstract commutative algebra.

5

Realising Hilbert’s programme in commutative algebra

The partial realisation of Hilbert’s programme in commutative algebra initiated by Coquand and Lombardi is clearly of interest to any one—mathematician, logician, philosopher, or 35

The references that occur in this and in the following quotes are adapted to the present bibliography.

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otherwise—who is concerned with the foundations of mathematics. For this kind of addressee we will therefore try to shed some light on the recent developments in that area, actually on a particular angle which we believe to be sufficiently representative. As this must anyhow fall short of a comprehensive treatment that takes into account all technical details, we will rather focus on a certain line of concepts and results, and leave it at an overview that will hardly satisfy any working expert but—as we hope—will be of some use for the general reader. More detailed introductions and surveys are available [26, 30, 86], to which our few pages can hardly be more than an appetiser. As compared with constructive algebra along the lines of Bishop [90] and Kronecker [45], extensive use is made of methods from point–free and formal topology [76, 83, 114] and in particular of distributive lattices [7, 75, 133]. Among the many general monographs on commutative algebra as done within customary mathematics we refer to [4] for a concise classic, and to [46] for a more recent and comprehensive source.

5.1

Rings and ideals

The basic concept of commutative algebra is the one of a commutative ring, which we briefly recall first. A ring (with unit36 ) is a set R with two distinguished elements, zero 0 and unit 1, and two binary operations, addition + and multiplication ×, such that the following conditions are satisfied: (R, +, 0) is an Abelian group;37 (R, ×, 1) is a monoid; and × distributes over + from both sides. A ring is commutative if so is multiplication, which of course is commonly denoted by juxtaposition. There is a ring with only one element, which a fortiori is commutative: that is, the trivial ring, usually written as 0, in which 1 = 0. A commutative ring R with 1 6= 0 is an integral domain if for every pair a, b ∈ R with ab = 0 either a = 0 or b = 0; and a field if for every a ∈ R either a = 0 or else ab = 1 for some b ∈ R.38 Clearly, every field is an integral domain. The prime examples of an integral domain and a field are the sets Z and Q of the integral numbers and the rational numbers, respectively—of course endowed with the usual operations of addition and multiplication, and with the distinguished elements 0 and 1 that are commonly denoted as such. Also, the ring of polynomials K [T ] in the indeterminate T with coefficients in a field K is an integral domain. But what are, in commutative algebra, the ideal objects in Hilbert’s sense? The prime example is presumably the concept of an ideal that Dedekind has made of Kummer’s “ideal numbers” [44, 43, 42, 41]. In modern terminology, an ideal a of a commutative ring R is an additive subgroup that is closed under multiplication by arbitrary ring elements, which is to say that a is a subset of R which satisfies the following three conditions: 0 ∈ a,

a ∈ a ∧ b ∈ a ⇒ a + b ∈ a,

a ∈ a ∨ b ∈ a ⇒ ab ∈ a .

(4)

An ideal a equals the ring R precisely when 1 ∈ a; the zero ideal is usually denoted by its only element 0. These two ideals are trivial examples of a principal ideal Ra = {ra : r ∈ R} generated by a single element a of R, which is often denoted by (a) or the like. An ideal p of a commutative ring R is a prime ideal if p 6= R, and if the converse of the last condition in (4) holds too: that is, if 1 ∈ / p, and if for every pair a, b ∈ R with ab ∈ p 36

It is nowadays common to suppose that a ring has a unit. A group is a monoid in which every element has an inverse. 38 A field in this sense is a discrete field in the terminology of [90]: that is, one can decide for each element a whether a = 0. 37

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either a ∈ p or b ∈ p. A maximal ideal of R is an ideal m with 1 ∈ / m such that for every a ∈ R either a ∈ m or else 1 − ab ∈ m for some b ∈ R. Every maximal ideal is a prime ideal. The zero ideal 0 is a maximal ideal (respectively, a prime ideal) precisely when R is a field (respectively, an integral domain). In particular, if R is a field, then 0 is the only ideal 6= R. The prime examples of a prime ideal are the principal ideals (p) of Z that are generated by the prime numbers p, which are all maximal ideals of Z. In K [T ] for any field K a principal ideal generated by an irreducible polynomial is a maximal ideal. Apart from 0, there are no other prime ideals of Z and K [T ]—at least in classical algebra.39 It is noteworthy that the definitions above of a prime ideal and of a maximal ideal, just as the one of an ideal, only require to quantify over the elements of the given set R. To quantify over the ideals of R is however necessary for the alternative definition of a maximal ideal which the name of this concept comes from: as an ideal m which is maximal among the ideals 6= R, which is to say that, first, m 6= R and, secondly, that for every ideal a 6= R if m ⊆ a, then m = a.

5.2

Noetherian rings

As we have just seen, ideals in general are ideal objects inasmuch as they are of the next higher type. In fact, ideals of a ring R are substructures of the algebraic structure of R: that is, subsets of the set R distinguished by a certain set of properties. To quantify over the ideals of R therefore means to quantify over a subclass of the power class of the set R. Admittedly, this nature of a fairly generic subset is hardly visible for the most common kinds of ideals, which indeed are concrete numbers rather than ideal entities. For instance, a principal ideal Ra = {ra : r ∈ R} can be represented by any generator a. More generally, a finitely generated ideal is of the form Ra1 + · · · + Rak = {r1 a1 + . . . + rk ak : r1 , . . . , rk ∈ R} with a1 , . . . , ar ∈ R, and thus can be identified with any finite list (a1 , . . . , ar ) of generators. In particular, a finitely generated ideal still lives on the same level as the ring elements. It is a lucky coincidence that this identification corresponds to the common habit of writing a = (a1 , . . . , ar ) or similarly in place of a = Ra1 + · · · + Rak . But aren’t many ideals finitely generated anyway, or even principal? The answer is that it depends, as follows, on whether one chooses to work in classical or in constructive algebra. The principal ideal rings are the rings of which every ideal is principal. Clearly every field is a principal ideal ring. In classical algebra quite a few other rings are principal ideal rings, as there are Z and K [T ] where K is a field. The Noetherian rings—according to the one of the several classically equivalent definitions that goes back to Hilbert—are the rings of which every ideal is finitely generated. Clearly every principal ideal ring is Noetherian. By the Hilbert basis theorem the ring K [T1 , . . . , Tn ] is Noetherian which consists of the polynomials in n > 1 indeterminates T1 , . . . , Tn with coefficients in a field K. This polynomial ring K [T1 , . . . , Tn ] is not a principal ideal ring unless n = 1. In constructive algebra one can still prove [90], by means of the Euclidean algorithm to compute the greatest common denominator of two integers, that an ideal of Z is principal 39

As we shall explain below, in constructive algebra this remains valid provided that one focusses on the finitely generated ideals.

20

provided that it is finitely generated. A similar argument applies to K [T ] where K is a field as defined above, i.e. a nontrivial ring with a zero test, by which assumption one can determine the degree of any given polynomial, which is to be used for K [T ] in the same way as the modulus of an integer is for Z. One cannot expect however to have a constructive proof that any of these examples is a Noetherian ring according to the Hilbertian definition, or even a principal ideal ring: i.e., that every ideal is finitely generated, let alone principal. As it was put in [90, p. 193], [. . . ] if by Noetherian we mean that every ideal is finitely generated, [. . . ] [then] only trivial rings are Noetherian in this [Hilbert’s] sense from a constructive point of view. In fact, a certain fragment of the law of excluded middle is necessary already for that the two– element field F2 = {0, 1} be Noetherian in Hilbert’s sense. The two most popular classical equivalents of Hilbert’s concept are equally problematic: Noether’s own definition that every ascending chain of ideals is eventually constant; and Artin’s variant that every nonempty set of ideals has a maximal element. All this will be made more precise in the appendix. In constructive algebra it therefore is useless to speak of a Noetherian ring in any of those particular senses, let alone of a principal ideal ring. Yet why did Seidenberg ask “What is Noetherian?” [118]. The reason is that there indeed are classically equivalent but still constructively meaningful variants of Noether’s ascending chain condition. The most popular one, put forward independently by Richman [107] and Seidenberg [118], says that in every ascending chain of finitely generated ideals at least two successive terms are equal. In constructive algebra many rings are Noetherian in this sense [90], including the polynomial ring K[T1 , . . . , Tn ] over a field; see also [97, 98, 117, 129]. Also, to prove with constructive means the termination of Buchberger’s algorithm for the computation of Gr¨obner bases, which is one of the cornerstones of computer algebra [1], it suffices that any K[T1 , . . . , Tn ] is Noetherian ` a la Richman and Seidenberg [96]. Another constructively relevant variant of the concept of a Noetherian ring consists of the rings on which one can do Noetherian induction [74, 96, 99]. This classical contrapositive of Artin’s aforementioned maximal principle says that if a property P (a) of finitely generated ideals a of a given ring R is hereditary, then P (a) holds for every finitely generated ideal a of R, where “hereditary” means that, for every a, if P (a0 ) for every a0 % a, then P (a).40 This however requires a generalised inductive definition, which goes beyond first–order logic [30]. In constructive algebra, by the way, Noetherian induction does not suffice for proving that every nontrivial ring has a maximal ideal, and thus a prime ideal. In classical algebra this is a particular case of Artin’s maximal principle mentioned before, which enables one to avoid the use of Zorn’s lemma in the context of Noetherian rings. Without the hypothesis that the ring is Noetherian, existential statements of this kind have been classified as forms of the axiom of choice; we refer the reader to [8] for more details including the original references. Yet being classically equivalent to the classical understanding of a Noetherian ring, both the Richman–Seidenberg concept and the one based on Noetherian induction only make use of quantification over finitely generated ideals. In spite of—or rather just because of—this seeming restriction, each of these concepts still includes most of the concrete examples that the working algebraist encounters in everyday practice, and then makes possible to give constructive proofs of theorems that are of practical interest. 40

In other words, the reverse inclusion order ⊇ is progressive.

21

Likewise, many theorems of constructive algebra are formulated for finitely generated ideals only; examples will be given later on. Perhaps the most fundamental reason for focussing on finitely generated ideals is that they can be managed more effectively, in both mathematical and programming practice: within a definition or a theorem, and as input or output of an algorithm or a computer program. More pithily put, in general one can only put one’s hand on an ideal if it is finitely generated. Working with the Richman–Seidenberg concept of a Noetherian ring sometimes requires us to add the precondition that the ring under consideration be coherent. This is a closure condition that in these cases allows us to carry out a proof of a theorem about finitely generated ideals without having to leave the realm of finitely generated ideals. A commutative ring is coherent [90] if, first, the intersection of any two finitely generated ideals is finitely generated and, secondly, if the annihilator (0 : a) = {x ∈ R : ∀a ∈ a (ax = 0)} of any finitely generated ideal a is finitely generated. In particular, the finitely generated ideals are closed under forming finite intersections, and thus under forming transporter ideals: (b : a) = {x ∈ R : ∀a ∈ a (ax ∈ b)} . Clearly, coherence would be automatic with the Hilbertian concept of a Noetherian ring; whence to assume coherence would be redundant for Noetherian rings in classical algebra.41

5.3

Spectra and schemes as distributive lattices

It often happens in (classical) commutative algebra [46] that the Zariski spectrum Spec(R) of a commutative ring R occurs in the course of a proof. The points of this space are the prime ideals of R: that is, as we have already recalled, the subsets p of R which satisfy 1 6∈ p , ab ∈ p ⇔ a ∈ p ∨ b ∈ p , 0 ∈ p, a ∈ p ∧ b ∈ p ⇒ a + b ∈ p,

(5)

where a, b ∈ R. Moreover, Spec(R) is endowed with the Zariski topology whose open sets of points—henceforth “opens”—are usually given as the complements of the closed subsets. The latter are the subsets Z (a) = {p ∈ Spec (R) : a ⊆ p} where a is an arbitrary ideal of R. The closure of a point p of Spec (R) consists of the points q which, as prime ideals and thus as subsets of R, lie above p: that is, q ⊇ p. Hence the closed points of Spec (R) are precisely the points of the subspace Max (R) that consists of the maximal ideals of R. In particular, if R is an integral domain, then not every point of Spec (R) is closed unless R is a field; the zero ideal 0 even is the only generic point of Spec (R): that is, the closure of this point is the whole space. To give an example, the initial part of Spec (Z) looks like (2)

(3) (5) - ↑ % ... 0

(7)

(11) . . .

41 Coherence however is of interest for non–Noetherian rings in classical commutative algebra [62, 63] and is made use of in the theory of control systems, see e.g. [100].

22

where → denotes inclusion ⊆. In other words, 0 → (p) also means that the point (p) is in the closure of the generic point 0. It is not just for convenience of notation that the closed subsets of Spec (R) are given conceptual priority over the open subsets: they form, via their traces on the subspace Max (R), a vast generalisation of the time-honoured concept of an algebraic variety, i.e. the set of common roots of finitely many polynomials. The latter occurs in the special case in which R = K [T1 , . . . , Tn ] where K is an algebraically closed field, such as the field of algebraic numbers or the field of complex numbers. In this case, and by the Hilbert Nullstellensatz, the maximal ideals of K [T1 , . . . , Tn ] are precisely the finitely generated ideals (T1 − t1 , . . . , Tn − tn ) with t1 , . . . , tn ∈ K; whence Max (K [T1 , . . . , Tn ]) ∼ = Kn . In other words, the maximal ideals of K [T1 , . . . , Tn ] correspond to the points of n–dimensional affine space over K. Any such interpretation of course requires us to identify, following Descartes’ algebraisation of geometry, a point of this space with the n–tuple t = (t1 , . . . , tn ) of its coordinates t1 , . . . , tn ∈ K. More specifically, Z (a) ∩ Max (K [T1 , . . . , Tn ]) ∼ = {t ∈ K n : h1 (t) = . . . = hr (t) = 0} for every ideal a = (h1 , . . . , hr ) of K [T1 , . . . , Tn ]. In other words, the closed subsets of Max (K [T1 , . . . , Tn ]) correspond to the algebraic varieties in Kn . This strong link to classic algebraic geometry notwithstanding, admitting non-closed points is essential for the Zariski spectrum’s pivotal role in bridging the gap between algebraic geometry and algebraic number theory: in both fields the results of commutative algebra are used extensively. In fact one had to go for (and beyond) a generalisation of the Zariski spectrum, for Grothendieck’s concept of a scheme, to which we will return later. For the time being we leave it at an example whose relevance should be clear in view of what we have recalled above. This example is AnK = Spec (K[T1 , . . . , Tn ]) , the affine scheme of dimension n over the field K. We need to stress again that the points of the topological space Spec(R), the prime ideals of the commutative ring R, are subsets of the given set R. As a consequence, the opens of Spec(R) are—as particular sets of prime ideals—even sets of subsets of R. If one seeks to keep the type level as low as possible, then one cannot possibly stick to this definition of the topology on Spec(R). Instead, one needs to deal with this space of points in terms of point-free topology [76, 75], in which the received conceptual priority of points over opens is reverted: that is, the opens rather than the points are seen as the primitive objects. As we shall see below, one can even go one step further and work with appropriate indices, or names, for the elements of a basis of opens; in fact the elements of R are perfectly suited as indices of this kind [60, 115, 116, 121]. By so doing one also follows the lines of what nowadays is known as formal topology [83, 114]. A point is then identified with a neighbourhood filter: that is, a collection of opens each of which contains the point. Of course any such neighbourhood filter needs to be rich enough so that it locates the point in a sufficiently precise way; whence the ontological status of the point can be neglected: The points (prime ideals, . . . ) constitute powerful intuitive help, but they are used here only as suggestive means with no actual existence. [26] 23

In view of the complexity of points and opens for the Zariski spectrum noticed above, at first glance the move from points to opens seems to make the situation even worse. On the contrary, it gives the chance to do much better: the elements of R can be used to index enough opens of Spec(R) (see below for more details); whence they can be used as finite approximations of points in much the same way in which, say, finite decimal fractions approximate real numbers. 5.3.1

Spectral spaces

The chief reason why, for topological methods in commutative algebra, one can get by on point-free topology is that “most of the topological spaces introduced in commutative algebra are spectral spaces” [26], with the space of minimal prime ideals (see below) being an exception that confirms the rule. In particular, Spec(R) is a spectral space, which indeed is the paradigmatic example of a spectral space: “these [the spectral spaces] are precisely the spaces that arise as Zariski spectra of commutative rings” [133, p. 121].42 But what does it mean that Spec(R) is a spectral space, and what is this good for? A spectral space is a topological space X which is sober (that is, every nonempty irreducible closed subset of X has a unique generic point—or, equivalently, X is homeomorphic to the space of completely prime filters of the frame of opens of X), and whose compact opens form a basis K (X) of the topology that is closed by finite intersection. In particular, a spectral space X is a compact Kolmogoroff (or T0 ) space, and K (X) is a distributive lattice. Note that every Hausdorff space is sober, but not every spectral space is Hausdorff. A spectral mapping is a continuous mapping F : X1 → X2 with the property V ∈ K (X2 ) ⇒ F −1 (V ) ∈ K (X1 ) , by which it induces a homomorphism F −1 : K (X2 ) → K (X1 ) between the distributive lattices. A key feature of spectral spaces is that the category of spectral spaces with spectral mappings is equivalent to the category of distributive lattices with lattice homomorphisms [9]. Clearly one assigns any spectral space X to the distributive lattice K (X), and any distributive lattice L to the spectral space Pt (L) whose points are the prime filters of L and which inherits its topology from L (for technical details see the appendix). Categorical equivalence then means X∼ = Pt (K (X))

and K (Pt (L)) ∼ =L

in which the isomorphisms are compatible with continuous mappings in X and lattice homomorphisms in L, respectively. In other words, one can go back and forth from spectral spaces to distributive lattices in a natural way, and without loosing any information. In particular, the compact opens of Spec(R) form a distributive lattice whose prime filters correspond to the prime filters of the ring R and thus to the prime ideals of R. With classical logic the prime filters are indeed the complements of the prime ideals (see again Appendix 6.3), but with intuitionistic logic the former are to be given priority over the latter: [. . . ] because it is at these objects [the prime filters of R rather than its prime ideals] that we wish to localise, and since ¬¬ = 6 id, we must deal with them directly [130, p. 194]. 42

For proofs of this correspondence in point-set and point-free topology see [70] and [9], respectively.

24

5.3.2

Joyal’s lattice

It seems as if one would not have gained much by replacing the Zariski topology on Spec(R) by the distributive lattice of its compact opens. Although this lattice only contains the compact opens rather than all the opens, it still consists of sets of points of Spec (R): that is, sets of subsets of R. But, once arrived at this point, one can do better. The key observation is that a fairly natural basis of the Zariski topology consists of compact opens—thus also is a basis of the associated lattice—and is even indexed by the ring elements. The elements of this truly versatile basis are D (a) = {p ∈ Spec (R) : a ∈ / p} , with a ∈ R, which are the complements of the closed subsets Z (a) for the principal ideals a = (a). Incidentally, the D (a) with a ∈ R form an affine cover of Spec (R): that is, each D (a) is itself isomorphic to a Zariski spectrum. In fact, D (a) ∼ = Spec (R[1/a]) where R[1/a] denotes the ring of (formal) fractions x/an with x ∈ R and n > 0. In view of the characteristic properties (5) of a prime ideal, the D (a) with a ∈ R satisfy D(1) = 1 , D(ab) = D(a) ∩ D(b) , D(0) = 0 , D(a + b) ⊆ D(a) ∪ D(b) .

(6)

The multiplicative ones among these properties (the top line of the above) ensure that this basis is closed under finite intersections; whence it indeed is a basis rather than just a subbasis. In particular, a generic compact open of Spec (R) is a finite union of basis elements: that is, D(a1 , . . . , an ) = D (a1 ) ∪ . . . ∪ D (an )

(7)

with a1 , . . . , an ∈ R. In view of all this, K (Spec (R)) is isomorphic to a lattice that can be defined in an elementary way: to the distributive lattice L (R) that Joyal [77] has given in terms of generators and relations (see also, for instance, [9, 75]). The generators of L (R) are the symbolic expressions D (a) indexed by the a ∈ R, and they are subject to the so-called support relations: D(1) = 1 , D(ab) = D(a) ∧ D(b) , D(0) = 0 , D(a + b) 6 D(a) ∨ D(b) .

(8)

Of course the support relations do not come out of the blue: they reflect the relations (6) between the compact opens, which in turn go back to the characteristic properties (5) of a prime ideal. The multiplicative ones among the support relations—i.e. the first line of (8)—allow for a simpler notation. First, the generators of L (R) are closed under finite meets; whence they form a basis of L (R) rather than just a subbasis; in other words, a generic element of L (R) is—in compliance with (7)—a finite join of generators D(a1 , . . . , an ) = D (a1 ) ∨ . . . ∨ D (an )

(9)

rather than a finite join of finite meets of generators. Secondly, while the order relation in an arbitrary distributive lattice has the normal form x1 ∧ . . . ∧ xm 6 y1 ∨ . . . ∨ yn , in L (R) any instance of 6 can even be simplified to one of the form D (a) 6 D (b1 , . . . , bn ), simply because in L (R) we have D (a1 · · · am ) = D (a1 ) ∧ . . . ∧ D (am ) . 25

Virtues of Joyal’s lattice The distributive lattice L (R) is isomorphic to K (Spec (R)), with D (a1 , . . . , an ) ∈ L (R) corresponding to D (a1 , . . . , an ) ∈ K (Spec (R)) for any a1 , . . . , an ∈ R. In particular, the prime filters of L (R) equally correspond to the prime ideals of R. But what is the advantage of L (R) as compared with K (Spec (R))? Each of these two distributive lattices has a basis whose elements are indexed by the ring elements, and thus live at the very type level of the elements of the given set R; in both cases moreover an arbitrary element is indexed by a finite list of ring elements. So what? Isn’t K (Spec (R)) just good enough, and why should one bother at all about moving to L (R)? The point is that this lattice [Joyal’s lattice L (R)] is constructively definable from [the ring R] [. . . ], so that we can bypass higher-order and irrelevant notions like the set of prime ideals [of the ring R] [. . . ] [136]. Of course one could view the symbols D (a1 , . . . , an ) for the basic opens as names for the elements of K (Spec (R)) and exclusively work with those names, neglecting the higher-order objects they denote. By so doing however one would end up with working in nothing but L (R). An impossible generalisation Why doesn’t Joyal’s method work more in general? More specifically, why cannot all open sets of a topological space, or even all subsets of a given set, be captured in a similar way?43 So, if Spec (R) allows for a symbolic representation in terms of the D (a) with a ∈ R, why shouldn’t this be possible for the whole power class P (R) in place of Spec (R)? Some evidence can be given with Cantor’s time-honoured argument for that P (R) has strictly bigger cardinality than R:44 if R is countable, then there are countably many D (a) with a ∈ R, which hardly suffice for any representation of P (R) unless R is finite. Needless to say, there are plenty of countably infinite rings; to mention only three of them, there is: the ring Z of integers, the field Q of rational numbers, and the field of algebraic numbers. Local-global principles With a point-free representation of the Zariski spectrum such as Joyal’s, also the local-global principles vital for commutative algebra need to, and can be put into concrete terms. A typical local-global principle says that a commutative ring R has a certain property E, for short E (R), already if E (Rp ) for every prime ideal p of R. Here Rp is the local ring of R at the point p, the ring of (formal) fractions x/s with x ∈ R and s ∈ R \ p. Note that one rather localises at the prime filter R \ p than at the corresponding prime ideal p. To avoid any such universal quantification over all the points of Spec(R), one has to reformulate it as a quantification over finitely many (compact, or basic) opens that form a covering. This can then be carried over to Joyal’s distributive lattice L (R) or expressed, via radical ideals, in terms of the given ring R. The practicability of this undertaking was demonstrated in various areas [82, 84], such as the one of the Serre conjecture including the theorems of Horrocks and Quillen-Suslin. Minimal primes It is noteworthy that no infinite (meets and) joins are required to describe L (R), so the lattice need not be complete—that is, have arbitrary joins—and coherent logic 43

With the discrete topology on an arbitrary set, for which every subset is open, the latter case is a special case of the former. 44 Those questions together with this answer have been communicated to us by Karl-Georg Niebergall.

26

is enough. However, for minimal prime ideals infinite disjunctions are indispensable, as they are licit in geometric logic. To have the minimal prime ideals of R as the prime filters of a distributive lattice [30], one indeed needs to allow for joins indexed by certain subsets of R: the topological subspace Min (R) of Spec (R) that consists of the minimal prime ideals corresponds to the quotient of L (R) which has the additional relations _ D (a) ∨ D (b) = 1 . b∈R:ab=0

To handle this one has to move from distributive lattices to frames (i.e., complete lattices in which the meet distributes over arbitrary joins), and from coherent to geometric logic. In particular, Min (R) is not a spectral subspace of Spec (R). 5.3.3

Radical ideals

The open subsets D (a) of Spec (R) correspond to the radical ideals √ a = {x ∈ R : ∃n > 1 (xn ∈ a)} of the ideals a of R, with D (a) ⊆ D (b) ⇔

√

a⊆

(10)

√ b.

This has prompted an alternative description of Joyal’s distributive lattice L (R) in terms of radical ideals. In fact, the formal Hilbert Nullstellensatz [28, 75] says that p (11) D (a) 6 D (b1 , . . . , bn ) ⇔ a ∈ (b1 , . . . , bn ) ; p whence L (R) is isomorphic to the distributive lattice of the (b1 , . . . , bn ) ordered by ⊆ with √ √ √ √ √ √ √ 0 = 0, 1 = R, a ∨ b = a + b, a ∧ b = a · b. The nontrivial direction ⇒ of (11) is a point–free substitute for \ √ a∈ {p ∈ Spec (R) : p ⊇ b} ⇒ a ∈ b ,

(12)

which by the way is needed to establish the aforementioned correspondence between open subsets of Spec (R) and radical ideals of R. Inasmuch as the prime ideals of R are the models of an appropriate theory (see the appendix), the implication (12) counts as a completeness theorem, of which the formal Hilbert Nullstellensatz (11) can be seen as a purely syntactical counterpart. This implication (12) moreover is tantamount to the special case in which a = 1 and b = 0, which is usually put in the form of its contrapositive 1 6= 0 ⇒ Spec (R) 6= ∅

(13)

that equivalently reads as “every nontrivial commutative ring has a prime ideal”. Under the name of Krull’s lemma, for arbitrary rings R this statement is known to be an equivalent of the Boolean ultrafilter theorem, a weak version of the axiom of choice [8], wheras for Noetherian rings R the implication (13) is provable with classical logic and without any form of the axiom of choice (Section 5.2). Note that no prime ideal at all occurs in the way (10) to put the radical ideal, which therefore is used in place of the alternative characterisation \ √ {p ∈ Spec (R) : p ⊇ b} = b , of which (12) is the nontrivial part, and which clearly violates the finite methods paradigm. 27

5.3.4

Generalisations

Projective spectra The point-free treatment of topological spaces for commutative algebra and algebraic geometry that began with Joyal’s lattice L(R) has already been extended beyond the case of Spec(R), first to the case of projective spectra. Before we outline this development, we briefly recall the concept of a projective spectrum. A graded ring is a commutative ring R such that a nonnegative integer, the degree deg (a), is assigned to every a ∈ R. These data are expected to satisfy, first, deg (ab) = deg (a) + deg (b) for every pair a, b ∈ R; and, secondly, R=

M

Rd

d>0

where Rd consists of the elements of R that are homogeneous of degree d: Rd = {a ∈ R : deg (a) = d} . P In particular, the elements of R are finite sums of the form ad with ad ∈ Rd . The prime example of a graded ring is the polynomial ring K[x0 , . . . , xn ], graded by the total degree, in n + 1 indeterminates x0 , . . . , xn with coefficients in a given field K. Following this example, it is fairly standard to assume for any graded ring R that R = R0 [x0 , . . . , xn ] for certain x0 , . . . , xn ∈ R1 . A homogeneous prime ideal of a graded ring R is a prime ideal p such that P ad ∈ p ⇔ ∀d (ad ∈ p) P for all ad ∈ R, and with ¬ (x0 ∈ p ∧ . . . ∧ xn ∈ p) .

(14)

(15)

As for Spec (R), all the D (a) = {p ∈ Proj (R) : a 6∈ p} with a ∈ Rd and d > 0 form basis of compact opens for the Zariski topology on Proj (R) = {p ⊆ R : p homogeneous prime ideal of R} , which is called the projective spectrum of R. In the aforementioned prime example, PnK = Proj (K[x0 , . . . , xn ]) is the projective scheme of dimension n over K. As for AnK , if the field K is algebraically closed, then the closed points of PnK correspond to the points of n–dimensional projective space over K. For any graded ring R as above, the projective spectrum Proj (R) too is a spectral space and thus can be represented in a point-free way by a distributive lattice [35]. This distributive 28

lattice P (R) is generated by the symbols D(a) with a ∈ Rd for some d > 0, which generators are subject to the relations D(a + b) 6 D(a) ∨ D(b) D(0) = 0 (16) D(ab) = D(a) ∧ D(b) D(x0 ) ∨ . . . ∨ D(xn ) = 1 for all a, b ∈ R. To ensure that D (a + b) is defined in R, one has to require for the first relation that a, b ∈ Rd for a common d > 0. Apart from this side condition, the first three relations of (16) are exactly as they are for Joyal’s lattice L (R); and P (R) is a point-free representation of Proj (R) in exactly the same way in which L (R) is one of Spec (R). When one puts the relations for P (R) in parallel to those imposed on the generators of L (R), one might miss D (1) = 1, which isP part of the W latter, from the former ones. In addition, one one might expect to encounter D ( ad ) = D (ad ), as it would perfectly mirror the additional condition (14) required from the homogeneous prime ideals. Both relations however are unnecessary—and in fact impossible to denote—in view of the restriction “a ∈ Rd for some d > 0” imposedPon the indices of the generators D (a), according to which one can neither P write down D ( ad ) nor D (1): in general ad is inhomogeneous, and 1 has degree 0. In fact D (1) = 1 has been replaced by the last relation D(x0 ) ∨ . . . ∨ D(xn ) = 1 of (16), which captures condition (15) and corresponds to the circumstance, well-known from classical geometry, that the n–dimensional projective space has n+1 affine pieces. This is also reflected by the fact that P (R) is isomorphic [35] to the result of glueing together all the distributive lattices     P (R) ∼ 1 =L R D (xi ) = 1 xi 0 with 0 6 i 6 n. Note that a quotient lattice of this type corresponds to an open subspace: to set u = 1 in an arbitrary lattice L means to focus on the lattice elements that are below u. Grothendieck schemes The case of the projective spectrum has hinted at a far more general point-free concept whose customary counterpart includes a large class of schemes `a la Grothendieck. As we have recalled earlier on, an affine scheme is one that is isomorphic to the Zariski spectrum of a commutative ring; more precisely the latter is to be viewed together with its natural structure sheaf of local rings. Now a (Grothendieck) scheme is a topological space endowed with a sheaf of local rings that is locally affine: that is, has an open cover consisting of affine schemes on each of which the given sheaf restricts to the structure sheaf [66, 48]. This clearly is reminiscent of the definition of a differentiable manifold as a topological space covered by homeomorphic copies of open pieces of Euclidean space; moreover the sheaf of local rings on a scheme is nothing but a generalised notion of a continuous function. It has turned out that the already fairly general notion of a Noetherian scheme can be based on distributive lattices rather than topological spaces; even more generally this can be done [36] for every scheme whose underlying topological space is spectral. Technically speaking, these particular schemes form a full subcategory that is equivalent to the category of distributive lattices enriched with appropriate sheaves of local rings. The resulting concept of a spectral scheme [36] instantiates the framework of formal geometries [116] given before in the context of formal topology [114]; it moreover generalises not only the distributive lattice—sketched above—which represents the projective spectrum

29

of a graded ring [35], but also the one which stands for the space of valuations of an abstract nonsingular curve [26]. With the latter lattice Dedekind and Weber’s time-honoured approach to Riemann surfaces via valuations, which already is of a point-free nature, has eventually—actually in the second-next century—been reduced to an appropriate low type level. It is noteworthy that most of the material required for linking [36] the concept of a spectral scheme with the one of a Grothendieck scheme has already been present since the early days of modern algebraic geometry [66, 1, 6.1]—of course formulated in terms of points rather than opens. Among other things, it then was known that the topological space of a Noetherian scheme X is spectral; and if Y is an arbitrary Grothendieck scheme, then the continuous part of every morphism of Grothendieck schemes X → Y is a spectral mapping. In this context it is therefore legitimate to repeat Rota’s question [113, p. 220]: What would have happened if topologies without points had been discovered before topologies with points, or if Grothendieck had known the theory of distributive lattices?

5.4

Krull dimension of rings and lattices

The Krull dimension of a commutative ring R is usually defined as the greatest possible length n of a chain of prime ideals p0 $ p1 $ . . . $ pn . For example, dim (Z) = 1, because 0 $ (p) for any prime number p is the longest possible chain of prime ideals in Z. Also, every field has Krull dimension 0, because 0 is the only prime ideal of a field. The Krull dimension of a commutative ring R is the special case X = Spec (R) of the Krull dimension of a topological space X: that is, the greatest possible length n of a chain of nonempty irreducible closed subsets X0 ' X1 ' . . . ' Xn . In fact there is an order-reverting correspondence between the prime ideals of R and the nonempty irreducible closed subsets of Spec (R): the latter are exactly the Z (p) with p a prime ideal of R. Even for n fixed, to define dim (R) = n in the customary way recalled above requires to quantify over the prime ideals of R. However, the topological nature of Krull dimension together with the representation of the Zariski spectrum by a distributive lattice have allowed for characterisations of Krull dimension that are completely elementary [28, 32, 33, 81, 86]. Still, each of these characterisations is, in classical algebra, equivalent to the customary definition recalled above; whence one can use the former to redefine the latter without any talk of prime ideals. The elementary characterisations can be traced back to the concept of Krull dimension for distributive lattices developed by Espa˜ nol [49, 50, 51] following Joyal. One of these characterisations of dim (R) involves directly the generators of the lattice L (R):

30

Lemma 2 For each n > 0 we have dim (R) 6 n precisely when for all a0 , . . . , an ∈ R there are b0 , . . . , bn ∈ R such that D (a0 ) ∧ D (b0 ) = 0 D (a1 ) ∧ D (b1 ) 6 D (a0 ) ∨ D (b0 ) .. .. .. . . . D (an ) ∧ D (bn ) 6 D (an−1 ) ∨ D (bn−1 ) 1 = D (an ) ∨ D (bn )

(17)

For instance, dim (R) 6 0 if and only if every generator of L (R) is complemented by another generator, in which case L (R) is a Boolean algebra. In view of the topological character of Krull dimension it is natural to involve the lattice L (R). However, this can be dispensed with, e.g. along the lines of the formal Hilbert Nullstellensatz (11), and another characterisation of Krull dimension can be put entirely in terms of the ring R. This in fact was given by Lombardi [81] before L(R) re–entered the stage. Lemma 3 For each n > 0 we have dim (R) 6 n precisely when for all a0 , . . . , an ∈ R there are b0 , . . . , bn ∈ R and k0 , . . . , kn ∈ N such that     ak00 ak11 · . . . · aknn (1 + an bn ) + . . . + a1 b1 + a0 b0 = 0 . (18) For example, dim (R) 6 0 if and only if ∀a ∈ R∃b ∈ R∃k > 1.ak (1 + ab) = 0 .

(19)

Note that if R is a reduced ring, then in (19) one can arrive at k = 1; whence in this case dim (R) 6 0 if and only if the commutative ring R is von-Neumann regular : that is, for every a ∈ R there is b ∈ R such that aba = a. Also, dim (R) 6 1 if and only if   ∀a0 ∈ R∀a1 ∈ R∃b0 ∈ R∃b1 ∈ R∃k0 > 1∃k1 > 1.ak0 ak1 (1 + a1 b1 ) + a0 b0 = 0 . Further there is an inductive characterisation of Krull dimension, which of course is particularly suited for proofs by induction. The key idea is to reduce the Krull dimension of a ring R to the Krull dimensions of the quotient rings R/Na of R modulo the boundary ideals √ Na = Ra + ( 0 : a) (20) with a ∈ R. More precisely we have the following [33]: Lemma 4 For each n > 0 we have dim (R) 6 n if and only if dim (R/Na ) 6 n − 1 for every a ∈ R, where the trivial ring 0 has Krull dimension −1. It is a freak of history that with Lemma 4 the Krull dimension can eventually, several generations of algebraists and topologists later, be seen as a particular instance of the concept of inductive dimension in topology that is due to Brouwer, Menger, and Urysohn. According to this time-honoured concept—and roughly speaking—the empty set is of dimension 6 −1, while for n > 0 a topological space has dimension 6 n if every point has a basis of neighbourhoods with boundaries of dimension 6 n − 1. In fact, Spec(R) = ∅ precisely when R = 0, 31

and Spec (R/Na ) viewed as a closed subset of Spec(R) is nothing but the boundary ∂D (a) of D (a); we refer to the appendix for a proof of this easy but little–mentioned fact. The inductive concept of dimension is about as intuitive as is Krull’s. For example, in a two-dimensional object such as a plane the longest chains of nonempty irreducible closed subsets are the ones of the sort plane–line–point; and neighbourhood bases can be formed by open discs, whose boundaries are circles. Yet inductive dimension seems to be inherently more elementary than the latter—at least for the Zariski spectrum, for which Lemma 4 provides an elementary characterisation of Krull dimension by way of nothing but inductive dimension. The reader may suspect, however, that the inductive characterisation of Krull dimension by Lemma 4 does not entirely comply with the finite-methods paradigm. In particular it √ is unclear a priori whether the transporter ideals ( 0 : a), and thus the boundary ideals Na , are (at least radicals of) finitely generated ideals. This is the case, however, under the hypotheses that the ring R isw Noetherian and coherent, which is frequently assumed in concrete applications of Krull dimension (see below). In fact, if R is of this kind, then L (R) is a Heyting algebra with implication √ D (a) → D (b) = D( b : a) for any (ideals generated by) finite lists a and b of elements of R, for which in particular the √ transporter ideal ( b : a) is finitely generated [28]. Apart from this, Lemma 4 may be seen as a fa¸con de parler : as a convenient method, tailor-made for proofs by induction, to encode dim (R) 6 n still without prime ideals but in the conceptual way that makes up the strength of modern algebra. Decoding is always possible; it is fairly immediate, and only requires some notational effort (Lemma 2 and Lemma 3).

5.5

Concrete applications of Krull dimension

With the one-line characterisation (Lemma 3) of Krull dimension, Coquand and Lombardi [28] first proved dim (K [T1 , . . . , Tn ]) = n for every field K and n > 0; in particular, AnK , the affine space of dimension n over K, has indeed Krull dimension n. The idea of this proof is not hard to explain if one admits some suggestive terminology: a complement of a sequence a0 , . . . , an in a ring R is a sequence b0 , . . . , bn in R that satisfies (18). Now, on the one hand, every sequence a0 , . . . , an in K [T1 , . . . , Tn ] is algebraically dependent over K (simply because it has one element too many) which can be shown to imply that a0 , . . . , an has a complement; whence dim (K [T1 , . . . , Tn ]) 6 n. On the other hand, the sequence T1 , . . . , Tn is algebraically independent over K and thus cannot have a complement, which rules out the possibility that dim (K [T1 , . . . , Tn ]) 6 n − 1. 5.5.1

Kronecker’s theorem under logical scrutiny

The inductive characterisation (Lemma 4) of Krull dimension has enabled Coquand [23] to give an elementary constructive proof of a time-honoured theorem of Kronecker’s: Theorem 5 (Kronecker) For each n > 0, if dim (R) 6 n, then for any given h1 , . . . , hm ∈ R with m arbitrary there are g1 , . . . , gn+1 ∈ R such that D (h1 , . . . , hm ) = D (g1 , . . . , gn+1 ) . 32

(21)

Coquand’s proof [23] is of course done by induction: if m 6 n + 1, then there is nothing to prove; if however m > n + 1, then any given set of m generators is transformed linearly into a set of m − 1 generators, to which the induction hypothesis applies. The required linear manipulations of the generators are possible thanks to the additional information which is at one’s disposal by dim (R) 6 n. Just as its principal hypothesis, the conclusion of Theorem 5 is of a point-free topological nature: (21) as it stands is an equation in L (R). As is not untypical for the gain of generality by point-free topological methods, Coquand was able to get by without the received hypothesis that the ring R be Noetherian. This assumption still stands, albeit implicitly, behind the “first modern proof” [47] of Kronecker’s theorem by van der Waerden [132], which, anyway, is rather a geometric imagination of an idea of proof. In his proof of Theorem 5 Coquand [23] works both in the ring R and in the lattice L (R). For the purpose of a logical analysis of Theorem 5 we thus establish a two–sorted language S of first-order predicate logic with equality that is an appropriate common extension of the languages R and L of rings with unit and bounded lattices, respectively: 1. S has a sort ρ for ring elements and a sort λ for lattice elements; 2. S has the customary function and relation symbols of R and L: that is, 0, 1, +, −, ×, = for rings;45 and 0, 1, ∨, ∧, 6, = for lattices;46 3. S has a unary function symbol D of type ρ → λ. We will only have to write down variables a, b, c, d, . . . of type ρ. With the convention (9) we assume par abus de langage that D may take on finite lists of type ρ as arguments. Now, if we fix numerals47 n and m, then the conclusion of Theorem 5 θm,n ≡ ∀h1 . . . ∀hm ∃g1 . . . ∃gn+1 .D (h1 , . . . , hm ) = D (g1 , . . . , gn+1 ) is a coherent sentence of S. Likewise, and again for a fixed numeral n, the hypotheses of Kronecker’s theorem form a coherent theory Γ ∪ ∆ ∪ Λ ∪ {κn } whose components are as follows: — Γ denotes the (equational) theory of commutative rings; — ∆ stands for the (equational) theory of distributive lattices; — Λ consists of the additional axioms for Joyal’s lattice: that is, Λ = {ι, ζ , ∀a∀b.π (a, b) ∧ σ (a, b)} where ι, ζ , π, and σ stand for the (atomic) support relations (8): ι ≡ D(1) = 1 , π (a, b) ≡ D(ab) = D(a) ∧ D(b) , ζ ≡ D(0) = 0 , σ (a, b) ≡ D(a + b) 6 D(a) ∨ D (b) ; 45

The function symbol − for subtraction is required for the theory of rings to be equational. We refrain from using different symbols—such as ⊥ and >—for the bottom and top element of a bounded lattice; we rather keep to the ones that are equally common for the zero 0 and the unit 1 in a ring with unit. Note also that equality can be defined in terms of order (by antisymmetry: x = y ≡ x 6 y ∧ y 6 x), and order in terms of equality and meet (x 6 y ≡ x ∧ y = x) or join (x 6 y ≡ x ∨ y = y). 47 For the sake of readability we do not follow the convention to use underlined letters m, n, etc. for numerals. 46

33

— κn is taken from Lemma 2 to express that the ring has Krull dimension 6 n: that is, ^ κn ≡ ∀a0 . . . ∀an ∃b0 . . . ∃bn .χ (a0 , b0 ) ∧ µ (ai , bi , ai−1 , bi−1 ) ∧ ν (an , bn ) 16i6n

where χ, µ and ν are shorthand for the (atomic) dimension relations (17): χ (a, b) ≡ D (a) ∧ D (b) = 0 µ (a, b, c, d) ≡ D (a) ∧ D (b) 6 D (c) ∨ D (d) ν (c, d) ≡ 1 = D (c) ∨ D (d) In all, Theorem 5 for fixed numerals n and m reads as Γ, ∆, Λ, κn ` θm,n . Note that in this theorem there is no occurrence of implication → or (logical) disjunction ∨; in particular there is no branching in the corresponding dynamical proof tree. Also, the existential quantifier ∃ can only be found in κn and θm,n . Alternatively, Theorem 5 can be expressed without any talk of lattices and with fewer hypotheses, which however requires us to use formulas which are geometric but fail to be coherent. To do this we define the language R0 to be the language R of rings enriched with infinite disjunctions indexed by elements of N, and with the usual function of exponentiation (of a ring element by an integer) as a defined symbol. Now any order relation in L (R) can p be put as D (a)0 6 D (b1 , . . . , bk ) or equivalently, by the formal Nullstellensatz, as a ∈ (b1 , . . . , bk ). In R this reads as _

∃c1 . . . ∃ck .a` = b1 c1 + . . . + bk ck

(22)

`∈N

and therefore is, for a fixed numeral k, a geometric formula but not coherent. One can modify accordingly the conclusion θm,n of Theorem 5 to get the formula p p θ0m,n ≡ ∀h1 . . . ∀hm ∃g1 . . . ∃gn+1 . (h1 , . . . , hm ) = (g1 , . . . , gn+1 ) of R0 , whose matrix is a finite conjunction of formulas of the form (22). To keep to R0 , moreover, the sentence κn needs to be replaced by the equivalent of dim (R) 6 n from Lemma 3:     _ `0 `1 0 `n κn ≡ ∀a0 . . . ∀an ∃b0 . . . ∃bn a0 a1 · . . . · an (1 + an bn ) + . . . + a1 b1 + a0 b0 = 0 . `0 ,...,`n∈N

For fixed numerals n and m both κ0n and θ0m,n are geometric formulas of R0 but not coherent, and Theorem 5 reads as Γ, κ0n ` θ0m,n . 5.5.2

The theorem of Eisenbud–Evans and Storch

The inductive characterisation (Lemma 4) of Krull dimension has even allowed for a constructive proof [34] of the following [47, 125] (see also [80]):

34

Theorem 6 (Eisenbud-Evans, Storch) Let R be Noetherian, strongly discrete, and coherent. 48 For each n > 1, if dim (R) 6 n − 1, then for any given h1 , . . . , hm ∈ R[T ] with m arbitrary there are g1 , . . . , gn ∈ R[T ] such that D (h1 , . . . , hm ) = D (g1 , . . . , gn ) .

(23)

This constructive proof of [34] is done with the concept of a Noetherian ring given by Richman and Seidenberg. Accordingly, Theorem 6 as put above, following [34], is restricted to finitely generated input ideals (h1 , . . . , hm ), and to coherent rings. Of course neither of this moves was done in the classical proofs [47, 125]: while the former is clearly unnecessary for the Hilbertian concept of a Noetherian ring, the latter is automatic for these kinds of rings, which are provably coherent (Section 5.2). For the constructive proof of [34] one further needs to assume the classical tautology— which however is constructively valid for many rings that occur in the mathematical discourse [90]—that the ring is strongly discrete: that is, membership to any finitely generated ideal is a decidable predicate of the ring elements or, equivalently, the inclusion order between finitely generated ideals is decidable.49 By the formal Nullstellensatz again, the outcome (23) of Theorem 6 is tantamount to p p (h1 , . . . , hm ) = (g1 , . . . , gn ) . (24) If this is compared with the result (21) of Theorem 5 and its equivalent p p (h1 , . . . , hm ) = (g1 , . . . , gn+1 ) , the advance that was made with the step from Theorem 5 to Theorem 6 becomes clear once one looks at a particular case. In fact, if R = K[T1 , . . . , Tn−1 ] with K an algebraically closed field, then R has one variable less, but R[T ] has as many variables as K[T1 , . . . , Tn ]. Now Kronecker’s Theorem 5 says that every algebraic subset of the n–dimensional affine space Kn can be described by n + 1 polynomial equations, whereas according to Eisenbud–Evans and Storch’s Theorem 6 this can already be done with n polynomials. One thus can get by with one equation less than before; this n moreover is known to be the optimal bound. For instance, a point of the plane K2 cannot possibly be described by a single polynomial equation: rather, it needs to be seen as the intersection of two curves. Already in dimension 3, however, this issue had remained unsettled for quite a time, and “the history of these results is rather interesting” [47]: In 1891, 9 years after Kronecker [in 1882] had announced his theorem, Vahlen produced an example which, he claimed, showed that Kronecker’s result was the best possible. The example he gave is a curve in complex projective 3–space which, he ‘showed’, is not the intersection of 3 hypersurfaces [. . . ]. Vahlen’s error seems to have gone undetected until 1942 [actually 1941], when Perron [. . . ] exhibited 3 hypersurfaces whose intersection is the curve in question. (The year [?] before, Van der Waerden [. . . ] had given the first modern proof of Kronecker’s theorem.) In 1961 Kneser [. . . ] showed that the existence of Perron’s hypersurfaces was not an accident by proving that every curve in 3-space is an intersection of 3 hypersurfaces. 48

These hypotheses will be briefly reviewed in Section 5.2. Note that only the first of these hypothesis is required classically and can be found in the original papers. 49 In [90] strong discreteness is expressed by “the ring has detachable ideals”.

35

Soon after Kneser’s achievement Forster [59] in 1964 raised the question whether n equations suffice in arbitrary dimension n, but it took nearly ten more years until Eisenbud–Evans and Storch independently settled the issue in general [47, 125]. Unlike the case of Theorem 5 discussed above, for Theorem 6 the hypothesis that the ring R is Noetherian seems to be indispensable for the arguments given in [34, 47]. The related but weaker hypothesis that the topological space Spec(R) is Noetherian is sufficient for the avenue followed in [125], which by the way gives more evidence for the topological nature of the theorems of Kronecker and Eisenbud–Evans–Storch. Theorem 6 allows for a logical analysis that is completely analogous to the one we have done before for Theorem 5; hence we do not need to do it. However, it is in order to briefly discuss the following question: What are the problems, from a constructive perspective, with the way in which Eisenbud and Evans [47] prove their theorem in classical mathematics? Needless to say, this proof rests upon the received definition of Krull dimension, with (chains of) prime ideals. But additional use of prime ideals is made at least twice. √ First, the initial case dim (R) = 0 is started by observing that the quotient ring R = R/ 0 can be written as a finite product of fields; whence R[T ] is a principal ideal ring. In particular, the image of the input ideal has a single generator g in R[T ], for which in this p (h1 , . . . , hm ) √ case one can show that (h1 , . . . , hm ) = g in R[T ] as required. All this is possible thanks to (a special case of) the Lasker–Noether decomposition available for Noetherian rings, by √ which 0 can be expressed as the intersection of finitely many (minimal) prime ideals: √ 0 = p1 ∩ . . . ∩ pk . (25) In fact, by the assumption dim (R) = 0 every pi is a maximal ideal—or, equivalently, R/pi is a field. By further removing redundancies one can achieve that the p1 , . . . , pk are relatively prime; whence by the Chinese remainder theorem one arrives at a product of fields √ R/ 0 ∼ = R/p1 × . . . × R/pk as required. The sweeping use of prime ideals notwithstanding, this construction can be constructivised whenever one imposes three additional conditions on R, each of which however is a classical tautology: 1. Suppose that R is strongly discrete (see above). 2. Assume that R has a strong primality test [96]: i.e., one can decide whether any given finitely generated ideal of R is a prime ideal, and if it is not, then one can certify this by way of a counterexample. 3. Understand “R is Noetherian” as the classically equivalent finite-depth property [99, 98]: i.e., every tree whose nodes are labelled by finitely generated ideals of R has finite depth provided that along every branch of the tree the ideals labelling the nodes form an ascending sequence.50 Under these√additional hypotheses one can indeed [96, 98] grow a binary tree with root labelled by 0 which is labelled strictly increasingly by finitely generated ideals—and which therefore is finite—and whose leaves are labelled by finitely generated prime ideals p1 , . . . , pk as required in (25). 50

Alternatively, one can strengthen “R is Noetherian” to “R is strongly Noetherian” [96].

36

The second use of prime ideals in Eisenbud and Evans’s proof [47] at first glance seems to be more problematic. To prove the nontrivial part of an equation of type (24) they invoke the characterisation \ √ Spec (R) = 0 (26) of the nilradical

√

0 = {x ∈ R : ∃n > 1 (xn = 0)}

of R. The nontrivial part a∈

\

Spec (R) ⇒ a ∈

√

0

(27) (28)

of (26), which by the way is the case b = 0 of implication (12), however is of an essentially nonconstructive character. Richman has observed in a related context: P Theorem 3 [i.e., if ai X i has a multiplicative inverse in R[X], then ai is nilpotent for i > 1] admits an elegant proof upon observing that each ai with i > 1 must be in every prime ideal of R, and that the intersection of the prime ideals of R consists of the nilpotent elements of R. This proof gives no clue as to how to calculate n such that ani = 0, while such a calculation can be extracted from the proof that we present. [108] We would like to stress again that, as in Richman’s case, the invocation of (28) in Eisenbud and Evans’s proof [47] can be replaced by fully constructive arguments [34]. As we have recalled before, the implication (28) in general is a form of the axiom of choice that is typically proved by way of Zorn’s lemma, whereas in the present case of a Noetherian ring there is no need for any transfinite proof method provided that one has classical logic at hand (see Section 5.2). In particular, the proof of the Theorem of Eisenbud–Evans and Storch given in [47] works in Zermelo–Fraenkel set theory—without the axiom of choice but with classical logic—and most likely even in an appropriate fragment thereof. The conclusions both of Kronecker’s theorem and of the theorem of Eisenbud–Evans and Storch are typical examples of a statement whose input and output is of finite nature, and which therefore truly merits a deduction from its hypotheses that is exclusively done by finite methods. The constructive proof [34] of Theorem 6 does indeed contain, for n and m fixed with dim (R) 6 n − 1, an algorithm that transforms, in R[T ], any finite list h1 , . . . , hm of length m into a finite list g1 , . . . , gn of length n that satisfies (23). Just as Storch’s proof [125], however, this only covers the affine case, whereas Eisenbud and Evans [47] have done the projective case as well; see also [80]. A constructivisation of the projective case has been carried out very recently [112], with the representation recalled before [35] of the projective spectrum Proj (R) of a graded ring R by a distributive lattice P (R). To follow the case of Spec (R) treated in [34], the Krull dimension of P (R) had to be turned into an elementary characterisation of the graded Krull dimension of R.

5.6

Heitmann dimension: an exception that confirms the rule?

It is in order to conclude by a quick look into a nearby direction. The inductive definition of Krull dimension has prompted √ an inductive definition of the Heitmann dimension Hdim, for which in (20) the nilradical 0 is replaced by the Jacobson radical JR = {x ∈ R : ∀y ∈ R∃z ∈ R.(1 + xy)z = 1}. 37

(29)

As for the nilradical, in constructive algebra this definition of JR is given priority over the characterisation \ Max (R) = JR (30) of JR as the intersection of all the maximal ideals of R, which of course is classically equivalent, again in general with Zorn’s lemma, to (29). The parallels between (27) and (29) on the one hand, and (26) and (30) on the other hand, are plain. This inductive definition of Heitmann dimension has eventually led to generalisations of the theorem of Forster–Swan to the case of non-Noetherian commutative rings, as well as to improvements upon the bounds relevant in this context [31, 32, 39]. This gain of knowledge, however, requires us to handle the Heitmann dimension with particular logical care as follows [30]. Already the explicit formula ∀a∃b∀y∃z. (1 + a (1 + ab) y) z = 1

(31)

for Hdim (R) 6 0 is a prenex formula with two alternations of quantifiers. Although it is a firstorder formula, it is by no means a geometric formula; hence Barr’s theorem—even without the axiom of choice as an assumption to be eliminated—does not apply at all.51 Still according to [30] there are other, proof-theoretic methods (Gentzen’s Hauptsatz, negative translation) by which an intuitionistic proof can be obtained from a first-order classical proof.52 Alternatively [30] the dependence of b on a and of z on a, y in (31) above can be expressed by Skolem functions f and g, respectively, with which (31) reads as (1 + a (1 + af (a)) y) g (a, y) = 1 . This move even leads to a coherent formula whose dynamical proof tree has no branching at all.

6

Appendix

6.1

Noetherian rings and excluded middle

Bishop’s Limited Principle of Omniscience (LPO) says that the disjunction ∃n (an = 1) ∨ ∀n (an = 0)

(32)

holds for every infinite sequence a0 , a1 , . . . of binary numbers.53 For completeness’s sake we now recall the well–known argument that LPO follows from the assumption that there is any nontrivial commutative ring R that is Noetherian in the classical sense. To this end we suppose that R is a commutative ring that has both of the following classically trivial properties: 1. R is a discrete ring: that is, a = 0 is decidable for each a ∈ R; 2. R is a nontrivial ring: that is, 1 6= 0 holds in R. 51

A related result of Ishihara’s [71] seems not to apply either. Incidentally, methods of this kind can be used for syntactic proofs of Barr’s theorem without choice; see [91] and [94], respectively. 53 In other words, LPO says that every Σ01 –formula is decidable. 52

38

With intuitionistic logic we will deduce LPO from any one of the following three conditions: 4. every ideal of R is finitely generated; 5. every inhabited set of finitely–generated ideals of R has a maximal element; 6. every ascending chain of finitely–generated ideals of R is eventually constant. (Note the restriction to finitely–generated ideals in conditions 5 and 6.) We first observe that condition 6 is clearly implied by condition 5, but also that condition 6 follows from condition 4. In fact, under condition 4 every ascending chain a0 ⊆ a1 ⊆ . . . of ideals of R is eventually constant, no matter whether the ak are all finitely generated. To see this consider the ideal generated by all the ak , viz. [ a= {ak : k > 0} . If this a is finitely generated, a = Rb0 + · · · + Rbm say, then for every j 6 m we pick k (j) > 0 such that bj ∈ ak(j) , for which a ⊆ ak(0) + . . . + ak(m) . If we set K = max{k (j) : j 6 m}, then we have ak(j) ⊆ aK for all j 6 m; whence a ⊆ aK and thus aK = aK+1 = . . . as desired. Now we deduce LPO from condition 6 in conjunction with conditions 1 and 2. To do so, let a0 , a1 , . . . ∈ {0, 1} be given. In view of condition 2 the binary numbers 0 and 1 can be viewed as elements of R. The finitely generated ideals ak = Ra0 + · · · + Rak with k > 0 form an ascending chain: that is, a0 ⊆ a1 ⊆ . . . By condition 6, there is L > 0 such that aL = aL+1 = . . . By condition 1, either ai 6= 0 for some i 6 L or else ai = 0 for every i 6 L. In the former case case we are done; in the latter case aL = 0 and thus ak = 0 for all k, which is to say that an = 0 for all n.

39

6.2

Boundaries of basic opens for the Zariski spectrum

In this section we describe the boundary ∂D (a) of the basis open subset D (a) of the Zariski spectrum Spec (R) of a commutative ring R, which of course is only an exercise in classical logic and topology. The reason why we do this in some detail is that we want to show how ∂D (a) is linked to the boundary ideal Na , which in fact defines ∂D (a) as a closed subset. In general, the boundary ∂S of a subset S of a topological space is defined as the closure S minus the interior S 0 , where S 0 equals S precisely when S is open. As for Spec (R), note first that the closure of D (a) is √ D (a) = Z( 0 : a) . In fact, the complement of D (a), the so-called pseudocomplement of D (a), is the union of √ all the D (b) for which D (a) ∩ D (b) = ∅ or, equivalently, ab ∈ 0; whence √ Spec (R) \ D (a) = D( 0 : a) . √ Since Z (a) is the complement of D (a), and Na = ( 0 : a) + (a), we have √ ∂D (a) = D (a) \ D (a) = Z( 0 : a) ∩ Z (a) = Z (Na ) .

6.3

Prime filters of rings and lattices

A prime filter ξ of a distributive lattice L and a prime filter π of a commutative ring R are subsets which satisfy the conditions below. Since with classical logic the defining conditions for a prime filter of R are the characteristic properties of the complement of a prime ideal of R, we have put them in parallel: ξ prime filter of lattice L

π prime filter of ring R

p prime ideal of ring R

x∨y ∈ξ ⇒x∈ξ∨y ∈ξ a+b∈π ⇒a∈π∨b∈π ¬ (0 ∈ ξ) ¬ (0 ∈ π) x ∈ ξ ∧ y ∈ ξ ⇔ x ∧ y ∈ ξ a ∈ π ∧ b ∈ π ⇔ ab ∈ π 1∈ξ 1∈π

a∈p∧b∈p⇒a+b∈p 0∈p ab ∈ p ⇔ a ∈ p ∨ b ∈ p ¬ (1 ∈ p)

The prime filters of a distributive lattice L form a topological space Pt (L) with the family Ξ (x) = {ξ ∈ Pt (L) : x ξ}

(x ∈ L)

as a basis of opens, where we use the fairly customary notation x ξ ≡ x ∈ ξ, whose choice we shall explain below. So, x ξ means that Ξ (x) is a neighbourhood of ξ. The topological space Spec (R) is presented by the distributive lattice L (R) inasmuch as, classically, the prime filters ξ of L (R) correspond to the prime ideals p of R, via D (a) ξ ! a ∈ /p for every a ∈ R. This even defines a homeomorphism Pt (L (R)) ∼ = Spec (R) . 40

More specifically, the prime filters ξ of L (R) correspond to the prime filters π of R via D (a) ξ ! a ∈ π for every a ∈ R, and the prime filters π of R correspond to the prime ideals p of R via a∈π ! a∈ /p for every a ∈ R. The defining conditions for a prime ideal of R are clearly reflected by the characteristic properties of the closed subsets of Spec (R): p prime ideal of R

behaviour of Z (·)

a∈p∧b∈p⇒a+b∈p 0∈p ab ∈ p ⇔ a ∈ p ∨ b ∈ p ¬ (1 ∈ p)

Z(a) ∩ Z(b) ⊆ Z (a + b) Z(0) = Spec (R) Z(ab) = Z(a) ∪ Z(b) Z(1) = ∅

Likewise, the defining conditions for a prime filter of R are reflected by the characteristic properties of the (standard) basic open subsets of Spec (R): π prime filter of ring R

behaviour of D (·)

a+b∈π ⇒a∈π∨b∈π ¬ (0 ∈ π) a ∈ π ∧ b ∈ π ⇔ ab ∈ π 1∈π

D(a + b) ⊆ D(a) ∪ D(b) D(0) = ∅ D(a) ∩ D(b) = D(ab) D(1) = Spec (R)

Alternatively, one can consider the models of the propositional theory T (R) whose atomic propositions are the D (a) with a ∈ R and whose axioms are the universal closures of the following formulas: D(a + b) → D(a) ∨ D(b) D(0) → ⊥ D(a) ∧ D(b) → D(ab) > → D(1) The models µ of T (R) correspond to the prime filters ξ of L (R) via µ |= D (a) ! D (a) ξ for every a ∈ R, which also explains the customary use of the symbol for prime filters. A thorough treatment of all this has been carried out in [28]. Acknowledgements Veronika K¨oberlein, Miriam Kertai, and Natalia Rabel, three former Diplom students of the second author, have contributed to this paper by asking the right questions. Discussions with and suggestions by Michael Detlefsen and Karl-Georg Niebergall have turned out most useful. Davide Rinaldi and Tobias Friedl were so kind as to have a look at the manuscript. The first author is grateful to Andrea Cantini for an inspiring discussion on the themes of this article, which took place at a very early stage in its preparation. Luca Bellotti gave very useful comments on a draft of this paper. Jesse Anne Tomalty proof– read the paper and gave useful suggestions. Last but not least both authors are grateful to Godehard Link for his infinite patience. 41

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