Genna! Systems. Vol. 20. pp. 115-!:!6. Reprints available d1rectl) from the publisher. Photocopying ..... that of a vassal, which is also reflected in the name "non-standard" natural numbers ... its pale reflection as found within Cantor's Set Theory.
«.: 1991 GNdon and Breach Science Publishers S.A .
Int. 1 . Genna! Systems. Vol. 20. pp. 115-!:!6 Reprints available d1rectl) from the publisher Photocopying permitted by licen~ only
Printed in lhe United Kingdom
THE PHILOSOPHICAL FOUNDATIONS OF AT.TRRNATIVR SRT THF.ORY* PETR VOPENKA Charles University, Faculty of Mathematics and Physics, Department of the Philosophy of Mathematics and Natural Sciences, Malostranske nam. 25, 118 00 Praha 1, Czechoslovakia This paper deals with philosophical fundamentals of Alternative Set Theory and its relation to classical mathematics. The key problem is a possibility to mathematize the phenomenon of horizon and the corresponding infinity. Serious criticism of classical set theory is given, especially in connection with its understanding of this phenomenon. The relation between classical and natural infinity (studied by AST) is presented and the difference between ~ST and Non-standard analysis is discussed . At the end. the fundamental notions of AST are briefly characterised. specifically that of sets, classes , semisets. and the extended universe. The encompassing of the vagueness (u nsharpness) phenomenon by AST. and the concepts of static and dynamic AST are al!'ltl discussed. INDEX TERMS :
Set theory, alternative set theory, non-standard analysis , infinity, class, semiset.
In the book Mathematics in the Alternative Set Theory, Leipzig, 1979, I overestimated the readers. I believed that the philosophy of Alternative Set Theory would be immediately obvious to all and I therefore concentrated more on demonstrating some techniques . However, the traditional mathematical thinking is so establishedeven in these times when physics is reevaluating its traditional thinking- that most mathematicians are not able to carry out a phenomenological critique of classical ideas, let alone realize its consequences. Therefore in this paper I shall attempt to sketch, at least briefly, the philosophical foundations of Alternative Set Theory. Classical mathematical ideas are now formalized in Cantor's Set Them)'. Only those structures are possible which have a model in this theory (Godel 's completeness theorem) and thus, in this way, Cantor's theory enclosed within itself the whole of mathematics . In mathematics there must not be anything ·which does not have a model in Cantor's Set Theory, thereby transcending or bypassing it. So, speaking rather loosely, mathematics (more precisely the philosophy of mathematics) equals Cantor's Set Theory (more precisely, its philosophy). Within this rigid framework are various directions criticizing classical mathematics; however, these essentially only restrict Cantor's Set Theory. Consequently , it is possible to raise the quite justified objection against these criticisms that they simply put various obstacles Jl1 its way in the form of prohibitions and thus stunt its development. Furthermore, even mathematics castrated thus in various ways, have models in Cantor's Set Theory. This objection is also sometimes brought up against Alternative Set Theory, namely by those mathematicians who have not understood (or were unable to fathom out for themselves) the basic principles of this theory. *Translated by Alena Yencovska. 115
116
PETR VOPENKA
HOW IS IT POSSIBLE THAT SOME RESULTS OF CLASSICAL INFINITE MATHEMATICS ARE APPLICABLE IN THE REAL WORLD AND OTHERS ARE NOT This question should be asked by every philosophically thinking mathematician. That is where we shall start to unfold our theme. Contemporary infinite mathematics is based on the..,flassical idea of in[initt,-so considering that we are concentrating mainly on the set theoretical approach to mathematics-on the classical idea of actuallv infinite sets. Nowadays however, the existence of such sets in the real world is, to say the least, dubious . Even if we conceded that such sets do exist in the real world , they are not those to which infinite mathematics apply their results. We apply the results of infinite mathematics to certain phenomena that accompany the appearances of large sets. By a "large set" we understand to mean a set that is finite from the classical point of view but that is not located, as a whole and along with each of its elements, in front of the horizon currently limiting the view of the given set. Here we understand "view" in much more than just a visual sense, that is, essentially as a grasping and holding of some field of phenomena. So, for example, we apply results about the classical topological continuum to the table standing in front of us; that is, to a certain phenomenon engendered by the set of molecules constituting this table. This phenomenon is indeed a phenomenon of a continuum , but not, however, that one which is studied by classical infinite mathematics . Likewise we apply the results from infinite probability theory to the set of all people, and similarly in a number of other cases. If we say that these applications are only approximate and in fact we only model the real situations in classical mathematics, we are not answering the above question but rather, only reformulating it: how is it possible that some real situations admit models-often relatively faithful-in classical infinite mathematics? If we are to be consistent, we have to acknowledge that infinity , in some form, is not only beyond them, but already present in large sets. If infinity is applicable to certain phenomena which show themselves on large sets, then it has to be already present in some form. If it was not there then we could not apply it or, in other words, the situation in question would not admit a model within infinite mathematics. We shall call this form of infinit~ natural infinity. To sum up, contemporary inffnite mathematics lias applications in the real world because it applies its results about classical infinity to natural infinity. As opposed to Cantor's Set Theory which deals with classical infinity, Alternative Set Theory studies this natural infinity. Not every result about classical infinity is applicable in the real world. This is true particularly of results about sets of large cardinality. It is unlikely that we should find a proper example in the real world to which it would be possible to apply some theorem, let us say, about sets of cardinality ~w 2 • We might attempt to create such an example artificially but in it we would hardly appreciate the full value of the theorem in question even though from the point of view of Cantor's Set Theory it was considerable . Moreover, if we try to pass this example off for an application of the theorem in question, we would at best encounter a tolerant smile. By the way, this is exactly from whence the antagonism stems, that which many mathematicians feel towards the set of large cardinality offered by Cantor's Set Theory. However, as far as the applicability is concerned, it is not only a question of sets of large cardinality. For example, no physicist today would agree that the paradoxical
118
PETR VOPENKA
world which enables us to gain insights, and these lead us to obtam non-trivial results, to choose definitions, eventually axioms, and so on. However, we do not see the classical world, but rather, the natural geometrical world. The latter differs from the former only by the fact that classical infinity is replaced everywhere by the corresponding form of natural infinity. To be sure, it was here that the natural geometrical world revealed itself first to us and only from it did we give the exposition of the classical geometrical world. We did this by replacing the form of natural infinity that presents itself in the natural geometrical world by the corresponding form of classical infinity, not vice versa.
HOW IS CLASSICAL INFINITY DERIVED FROM NATURAL INFINITY Since the basic and simplest classical infinity is the one which is associated with the set of natural numbers, we shall concentrate mainly on this example. As we have said before, the starting point for deriving the classical exposition of the set of natural numbers is a certain sequence of these numbers going on to the horizon. The following three characteristics of the horizon are now important for our theme . Firstl y, we do not understand the horizon as the boundary of the world, but as a boundary of our view. So the world continues even beyond the horizon. Secondly, the horizon is not some line drawn and fixed in the world but it moves depending on the view in question, specifically, on the degree of its sharpness. The further we manage to push the horizon, the sharper the view. Thirdl y, for a phenomenon situated in front of the horizon, the closer it is to the horizon, the less definite it is. These characteristics of the horizon present themselves on the sequence of natural numbers as initially grasped. In the beginning it is clear and distinct but the closer to the horizon the more obscure it becomes until it disappears from our view beyond the horizon. Furthermore, this sequence does not have a last element; if it did, the horizon would be fully determined by this last element. So on this sequence, natural infinity presents itself to us in its most basic form, namely as countable natural infinity. We derive the classical countable infinity from the above sequence by constantly sharpening our view, that is, by moving the horizon further and further. Thus we also sharpen and prolong the sequence. However, we do not prolong it by adding more elements one by one (which is not possible anyway, since it has no last element), but in jumps always to new, more removed, horizons. The outcome of this sharpening process is a complete sharpening through which countable classical infinity stabilizes as unchangable, definite and sharp. So we come to the classical set of natural numbers by a complete sharpening of that sequence of natural numbers which is proceeding to the horizon. From this sequence we also infer properties of the set of all natural numbers-along with the sequence, we also sharpen its properties. Some properties, however, are transferred directly, for example, that for each natural number there exists a larger one. Of course we derive basic properties of the classical set of natural numbers only from the sequence of natural numbers, that is, advancing towards the horizon and not from those situated beyond it. An illustrative example is the assertion according to which each non-empty part of the set of all natural numbers has a least element. However that part of the natural numbers situated beyond the horizon does not have a least element.
ALTER NATIVE SET THEORY
'~s!
~
119
As an aside, note that in a similar fashion we progress from the natural geometrical world to the classical one . Also in this case we move the horizon both depth-wise and distance-wise. The fact that the classical exposition of natural numbers has been derived as explained above is also confirmed by a number of contemporary results about the natural numbers and sets in general . So for example, non -standard models document the fact that the set of classical natural numbers is a complete sharpening of just that sequence of natural numbers that is advancing toward the horizon. For the classical natural numbers appear hereeven within classical mathematics- again in their original role , as a sequence going towards the horizon . It is not important that , under the influence of tradition, this horizon be understood as fixed and definite . A characteristic property of natural numbers is their continuation beyond the horizon. This oroperty has been often used in mathematics- the best known example being that of infinitesimal calculus . The classical exposition denied this property to natural numbers , but non-standard models retrieved it into the light. The explosive development of non-standard analysis can be recognized as a joyful celebration of its rediscovery. Similarly in some more delicate considerations of mathematical logic it is necessary to distinguish the mathematical and the meta-mathematical natural numbers , when the latter are shorter. This distinction is possible thanks to the fact that the sharpening of properties may be held back in comparison with the sharpening of natural numbers. The question arises which of these numbers are the genuine ones . The classical exposition prefers the meta-mathematical numbers , the mathematical ones being their prolongation. However, we could equally well declare the mathematical numbers to be the genuine ones, with the meta-mathematical numbers being only a sequence of natural numbers advancing towards some unimaginably far off horizon . Within the classical exposition of sets we would probably agree on the following: if a part of the set of natural numbers has the property that its intersection with any finite set is a finite set, then this part is a set itself. But now consider yourself where you draw the conviction from , that any part of natural numbers that is defined from such a set of natural numbers by a formula of predicate calculus again has the above mentioned property.
WHICH INFINITY IS STUDIED BY CLASSICAL MATHEMATICS It might appear that a question formulated like this already contains the answer. Naturally, classical mathematics studies classical infinity. But in truth , this is only an act of faith on the part of classical mathematics . We are not able to maintain the sharpening of that sequence of natural numbers which is going on to the horizon (that is , moving the horizon further according to our view of the natural numbers) for very long, so that eventually we continue it only in our minds. However, not even in our minds are we able to reach as far as classical natural numbers would require. At a certain moment we must make a radical step and think that this sharpening process has led to the complete sharpening. Of course, what we hold during this sharpening process is always natural infinity even though it belongs , case by case, to a different view. That last radical step is nothing
120
PETR VOPENKA
else but the decision not to sharpen any further. Even after we have made that decision we still hold only natural infinity, the difference being simply that we ignore the corresponding horizon; or to put it another way we impose upon this horizon a character other than its own and we understand it as impassable, fixed and definite. The impossibility of stepping over this horizon forced classical mathematics to extend the subject of its study in a different direction, namely through creating powersets. The unchangeability and definiteness of this horizon is inherited by the subsets of the set of natural numbers which justifies the extension of this direction. But in fact these power-sets also represent only natural infinity even though now rather muddled and, not least, dependent on the horizon. When we realize this, a question such as whether or not the continuum hypothesis holds loses some of its absoluteness . Even classical mathematics then studies natural infinity; however, it does so inappropriately. Classical mathematics is restricted by the accepted limitations, mainly by those inflicted on the horizon. The acceptance of the hypothesis that the sharpening process can lead to a complete sharpening does not extend the field of our study but rather to the contrary , restricts it. The study of situations where the sharpening process itself is essential is thus completely blocked . To put it briefly, the laws that govern classical infinity are nothing more than a drastic restriction of the laws that govern natural infinity. In spite of this interesting line of analysis of Cantor's Set Theory as suggested above, it is not the aim of this paper to continue in this direction. Already from what has been said thus far, it is perhaps sufficiently obvious that Cantor's Set Theory is a study of structures of a certain sort created through natural infinity and, consequently, that the whole of this theory is a part of Alternative Set Theory. As far as power is concerned, both these set theories are, to say the least, of equal power. There is an essential difference between these theories in appreciating their theorems. In Alternative Set Theory there is a place for theorems about sets of cardinality ~W' ' since it is possible to reconstruct the whole of Cantor's Set Theory there. There are many such reconstructions and essentially they depend on the horizon and on the choice of sets belonging to the power-set. However, these "Cantor's structures" are, from the point of view of Alternative Set Theory, something artificial; their meaning is, in fact, only historical. So to put it rather metaphorically, we are talking about theorems which would correspond in geometry either to results about, say, a 238-gon, or that would concern a geometry which limited itself only to the inside of a given circle and forbade considerations of its exterior. Even in such a geometry it is possible to define the set of line segments whose prolongations go through a single point lying outside this circle. Indeed, considering the effort we would have to employ, such a definition would be more appreciated than one which we could formulate using the exterior of the given circle . Naturally, this does not suggest that everything that has been achieved in Cantor's Set Theory has no value in Alternative Set Theory. Quite to the contrary, it is possible to consider large areas of mathematics as being based on Alternative Set Theory . However, results achieved in these areas have a slightly different interpretation: essentially, that of the interpretation associated with their applications . So, for example, when the real numbers are constructed in Alternative Set Theory it is possible to accept, almost literally, everything that has been achieved in the ~:, 8-analysis of real functions. Actually, this was true also at the time when Cantor's Set Theory became the world of mathematics . Then Cantor's theory opened for~:, 8-analysis the door to functional analysis. Similarly, Alternative Set Theory opens a door to it, but for a different area in which the €, 8-analysis can develop.
ALTERNATIVE SET THEORY
121
WHEREIN LIES THE DIFFERENCE BETWEEN NON-STANDARD ANALYSIS AND ALTERNATIVE SET THEORY The basic difference between these theories can be briefly characterized as follows: Alternative Set Theory is aware that it studies natural infinity whilst Non-standard Analysis, like the whole of classical infinite mathematics up to now, is not aware of it. Naturally, this predestines the directions of development of both of these theories. The formalization and modelling of various situations in Alternative Set Theory begins with the way in which these situations present themselves and from the possibilities they offer for treating them. On the other hand, Non-standard Analysis begins with the classical models of these situations and, in order to treat them, employs means which it gained by stepping over the horizon, that is, that horizon which was designated by classical mathematics as a fixed and impassable boundary of the natural numbers. Currently we observe that Non-standard Analysis is gradually being applied to those applications in which the standard natural numbers model the path to the horizon , and the non-standard ones model their prolongation beyond the horizon. As opposed to previous infinite mathematics , Non-standard Analysis here presents these applications in a somewhat half-hearted manner, almost shyly. The classical continuum had been presented withou t any hesitation as a suitable, even the most suitable, model of the natural continuum, whilst its extension in Non -standard Analysis is mostly considered to be only a suitable technical tool. To put it briefly, the relation of Non -standard Analysis to Cantor's Set Theory is that of a vassal, which is also reflected in the name "non-standard" natural numbers and so on. Of course, by this, Non-standard Analysis renders itself powerless and gives rise to a justified criticism that it only proves known theorems in a different way. When Non-standard Analysis is obscured in this way then, naturally, everything it can prove from Cantor's Set Theory using its own theorems can also be proved directly . As long as this master-vassal relationship lasts, Non-standard Analysis cannot use all its potential, which lies mainly in new formalizations of various situations and not in new proofs of classical theorems. However, even if Non-standard Analysis started to treat its non-standard and standard objects equally, and while modelling various situations inspired itself by natural infinity, the question would still remain as to whether it could master natural infinity fully, since the path through Cantor's Set Theory does not give any guarantees for it. Notice that the infinity studied in Cantor's Set Theory is only natural infinity drastically restricted, so the effort to fathom out from here all the possibilities offered by natural infinity is not justified by anything at all. Such a path is notably contorted and misleading. It is necessary to approach the study of natural infinity directly and not through its pale reflection as found within Cantor's Set Theory. Such a direct approach is what Alternative Set Theory attempts. WHAT IS THE CONNECTION OF NATURAL INFINITY WITH INDEFINITENESS Already in classical antiquity there was an awareness of the close connection between these two phenomena. lts "apeiron" was not a sharp and definite infinity but a dark
PETR VOPENKA
122
and indefinite ocean which surrounds the world and in many places penetrates it. By the complete sharpening of natural infinity this connection has been disrupted. However, it continued to manifest itself through applications of infinite mathematics. Natural infinity is an abstraction from path leading to the boundaries of indefinite and blurred phenomena . Since we understand various phenomena exactly through grasping their boundaries (compare lat. "definitio"), the study of natural infinity is also a possible foundation for the science of indefiniteness. We shall thus begin with indefiniteness understood as widely as possible. Since, however, Alternative Set Theory is a theory of sets, we shall concentrate particularly on the distribution of extensions of various properties towards their boundaries . From this point of view Alternative Set Theory is a study of blurred sets. As opposed to the so called "fuzzy set theory ," we shall examine them as truly indefinite, without an underlying definite structure. That basic form of natural infinity which led, after completing the sharpening process, to the basic classical infinity, countability has been abstracted from the path leading on to the horizon. Even though the horizon is the most prominent form of a boundary of an indefinite phenomenon and we encounter it in one way or another in almost all cases of indefiniteness, it is not at all obvious that all forms of natural infinity should be reducible to the basic one in some simple mechanical manner. On the other hand, it is useful in using it to attempt to describe as many forms of natural infinity as possible for it would enable us to carry over into Alternative Set Theory various techniques developed in classical infinite mathematics and thus, in some directions, to progress quickly .
WHAT OBJECTS ARE STUDIED BY ALTERNATIVE SET THEORY Alternative Set Theory works with objects of two sorts: sets and classes. Sets are definite, sharply defined, unchangable and finite from the classical point of view. For the universe of sets we accept the axioms of ZFt except that the axiom of infinity is replaced by its negation . Sets serve two purposes. On the one hand they play their own role, that is, they are sets from the universe of sets ; on the other hand, they stand for various fixed and unchangable objects. In the latter case we ignore the inner structure of their elements as is common in classical set theory. The natural numbers are defined in the universe of sets by von Neumann's method (for technical reasons only) . Classes represent indefinite clusters of objects. The universal class V is the class of all sets, and N denotes the class of all natural numbers. Even in these two cases we deal with classes, not sets. This is not only for formal reasons (because they are not sets from the universe of sets) but for factual reasons. So, for example, the class N represents a path to some vastly remote horizon, inaccessible for us but still such that the class N satisfies Peano's axioms. Here, of course, we include in the PA system+ only those axioms , the length of which falls in front of the studied (concrete) horizon represented by the class FN which is a proper initial segment of the class N. Not even the class N (and likewise the class V) is thus sharply defined. tZermelo-Fraenkel :!:Peano Arithmetic
ALTERNATIVE SET THEORY
123
We shall leave unanswered the question of whether the elements of the class N can be identified with those natural numbers which classical mathematics believes itself to be studying, that is, with some absolute natural numbers. Anyway, no one nowadays is able to answer this question. If we ever obtain an answer it will probably be negative since a positive answer transcends human abilities. We cannot foresee at all what we would see if we shifted the horizon in some genuinely qualitative way. We cannot exclude that in such a case the ideas we held hitherto would completely collapse; eventually the world could start to fall apart into apeiron. The assumption that whenever we move the horizon further we wou ld see essentially in the same way as we do now, is the least inventive one . It was just this assumption that delayed the entrance of non-euclidian geometries into mathematics . Hence, if classical infinite mathematics bases its meaning on the claim that it studies some absolute infinity, if that is what constitutes its nobility, then we deal here only with a non-critically accepted conjecture. An important role in Alternative Set Theory is played by classes (we call them semisers) which are part of sets. Above all, it is these classes that represent the phenomena which appear when we are viewing large sets. For technical reasons it is usefu l to also consider sets to be classes and talk about proper classes and proper semisets .
THE TRUNK AND BRANCHES OF THE EXTENDED UNIVERSE By the extended universe is understand to mean the collection of all classes. One of the basic principles of Alternative Set Theory is the so called principle of the unboundedness of the extended universe. According to this principle, every class that is possible , i.e. consistently conceivable, belongs to the extended universe. We have no right to prohibit the existence of any such class since at the very moment we conceive of it we have already brought it into being in a certain way; it exists in our minds . This principle documents that Alternative Set Theory is not at al l as plebeian as might appear at first sight. Quite on the contrary , this theory does not exclude any class whose existence is not excluded by pure reason alone. Even the paradoxical decomposition of the sphere has its place in Alternative Set Theory , the only question being its place. For not all classes hold equal status . Their classification from those which present themselves somehow spontaneously to those which come from the wildest fantasy is one of the tasks undertaken by Alternative Set Theory. We now briefly mention the most important types of classes. When accepting the principle of unboundedness of the extended universe, the compatibility of this collection of classes ceases to be tenable. For each of two classes X, Y with certain properties , each might be consistently conceivable whilst that might not be the case for both of them together. If we bring to being the class X then the class Y cannot exist and vice versa . Through the acceptance of the existence of the class X we thus enter a certain branch of the extended universe which does not contain the class Y; the class Y is in another branch. Consequently, it is not p ossible to actualize the extended universe, that is, to view it as if all its classes already existed . Understandably, we cannot axiomatize it either since such a system of axioms would be contradictory. We can however investigate any particular branch which we want to work in through the predicate calculus . If we talk in such cases about the extended universe we mean the branch in question .
124
PETR VOPENKA
All the branches share a common part which we call the trunk of the extended universe . First of all, all sets from the universe of sets belong to the trunk, as do those classes that present themselves to us somehow spontaneously (e.g. the class FN), the classes obtained from these by a mental construction (the paradoxical decomposition of the sphere) and naturally also those classes whose existence does not clash with the existence of other classes. Even within the trunk, then, there are classes of various types. A considerable part of the trunk is axiomatizable; we shall denote the corresponding axiomatic system by AST. The system AST relates to the trunk analogously to the way the system ZF relates to Cantor's Set Theory. Thus AST does not capture the whole of the trunk. For example, in Alternative Set Theory, according to Godel's .completeness theorem which states that every consistent theory has a model, also holds. Furthermore, this theorem holds even for such theories where the proof of a contradiction is not to be found before the horizon of the given view. An example of such a theory is ZF since a contradiction in it has not (yet) been found. So in Alternative Set Theory even the ZF theory has a model which belongs to the trunk. However, its existence in AST is not provable. Naturally, if a contradiction in ZF was found, that is, if we succeeded in shifting the horizon in such a manner, there would then be no model of ZF within the trunk in question any more. Up to now, research in Alternative Set Theory has taken place mainly in the AST system. The occasional transcending of the AST system concerned the trunk rather than the branches. This development of study is understandable; the development of Cantor's Set Theory in the initial stages did not step beyond the framework given by the axiomatic system ZF. On the other hand, these investigations have shown that the AST system is sufficient for the study of a number of rigid real situations on which we have been focusing our attention. Naturally, it does not mean that this system is now definitively established. We do not exclude the possibility that adding some other axiom to this basic axiomatic system might become desirable in the fu ture . After all, in classical set theory Zermelo's axiomatic system was extended to the Zermelo-Fraenkel system. Summing up briefly, Alternative Set Theory is not axiomatizable and in no way is it permissible to identify it with the AST axiomatic system. There are models . of the AST axiomatic theory within ZF; they are essentially certain non-standard models of the theory of finite sets . We cannot model the trunk of the extended universe in ZF since, if we could, there would exist in ZF, for example, a model of ZF plus the existence of inaccessible cardinals, and so on. It is possible, however, to model the trunk in Cantor's Set Theory because of Godel's completeness theorem which we acknowledge to be valid within Cantor's Set Theory. Understandably, the whole of the extended universe does not have a model in Cantor's Set Theory.
THE STATIC AND DYNA!'vliC ALTERNATIVE SET THEORY The trunk of the extended universe also contains the class FN which is a representative. of the path towards the horizon. A number of other classes situated not only in the branches but also in the trunk depend on the manner in which this class is situated within the class N . If we do not change the degree of sharpness of our view during our study, that is, if we do not move the horizon neither further nor closer,
ALTERNATIVE SET THEORY
125
then FN stands for the same class all the time. In such cases, we work in the static Alternative Set Theory. However, a characteristic feature of the horizon is precisely its mobility. If we take that into account, we must work in the dynamic Alternative Set Theory . One of its basic principles is that the horizon may be moved both further and nearer. Suppose, for example, in one view the class FN is X and in another it is Y. Assume that X is a proper subclass of Y. This means that the latter view is sharper. In this latter view the class X cannot be brought into existence. More precisely, if X could be brought into existence then it would mean the view becoming Jess sharp since we would suddenly see an indefinite part where all parts had been definite before and thus FN = Y would cease to-hold . Thus when passing from the view where FN = Y to the view where FN = X a considerable part of the trunk of the extended universe changes. Only the sets from the class V do not change even though some of them sink beyond the horizon . Thus, when the view changes, some classes come and other ones depart . For example, it is well known that when the view is sharpened, changes of shapes occur. A table seen through unaided eyes has a different shape from a table seen through a microscope. These changes of shapes and of various phenomena in general can be continuous or discontinuous . The continuity with which we deal here has a different character than the one studied classically or within static Alternative Set Theory. As an aside , notice that although a discovery of a contradiction in Peano Arithmetic is considered unlikely, Alternative Set Theory is insured even against such a case . Such a contradiction would only mean that the scope of the class FN is limited from above by some concrete natural number. Everything that has been proved in Alternative Set Theory where this number is not situated before the horizon (that is, does not belong to FN) continues to hold. Naturally, changes of views are due not only to changes in the class FN but also to the movement from one branch to another. Here again, changes of some classes can be continuous and of others discontinuous. Of course this is only a very brief sketch of a part of the area of dynamic Alternative Set Theory. There is no doubt that this area is important; for example, in connection with the quest of physics to plumb the depths of the subatomic universe . Dynamic Alternative Set Theory opens a field of study to which there is no entry from Cantor's theory. In its dynamism, Alternative Set Theory qualitatively transcends the framework of Cantor's theory. It is not possible to find models of dynamic Alternative Set Theory in Cantor's theory . Even if we assume that we can model particular branches of the extended universe and particular cases of sitings of the class FN, we cannot model changes of views. The natural numbers and, in general, sets from the class V do not change with the view since they represent objects not dependent on that view. In order for dynamic Alternative Set Theory to have a model in Cantor's theory, the natural numbers in the models of all branches would have to be the same. This is not possible in Cantor's Set Theory .
REFERENCES I. P. Yopi'nka, MaThema Tics in The AlrernaTive SeT Theory, Teubner-texte zur Mathematik , Leipzig, 1979. 2. P. Yopenka, InTroducTion To MaThemaTics in The AITernaTi,·e SeT Theory, Alfa, Bratislava, 1990 (in Slovak).
126
PETR VOPENKA
Prof. RNDr. Petr Vopenka, DrSc., graduated from Charles University in Prague where he also received a DrSc. degree in 1967. He became an associate professor there in 1965 and full professor in 1990. He is head of the department of philosophy of mathematics and natural sciences, and in 1990, became a Minister of Education, Youth and Physical Training of the Czech republic. He is the author of several tens of publications including four monographs. His main topics of interest are set theory, logic and foundations of mathematics. He is a leading force in the group developing alternative set theory.
