We think we proved the consistency of set theory (ZFC), a first order theory. ... 1.4 Axiom of Union For any X there exists a set Y = J X, the union of all elements.
Consistency of Set Theory from the consistency of NFU set theory C´esar J. Rodrigues CCTC, Department of Informatics Minho University Braga, Portugal CIDMA, Department of Mathematics Aveiro University Aveiro, Portugal
1 Introduction Abstract We think we proved the consistency of set theory (ZFC), a first order theory. The proof presented here simplifies the proof in [6]. There were two important decisions: 1. The presentation of set theory NFU which is consistent. 2. The relativization of second Godel Incompleteness Theorem, which has some implications in this type of proofs. The consistency of set theory means the consistency of mathematics, since set theory may be used as a foundation for mathematics.
Russel paradox in set theory introduced the need to develop mathematics, with the need to avoid paradoxes. To respond to this situation, many philosophies of mathematics appeared, which presented different strategies to resolve paradoxes and to define what was mathematics. Set theory had to evolve from the naive set theory (the set theory corresponding to the first develpments after its invention by Cantor) to axiomatized set theory, introduced by Zermelo with the intervention of Fraenkel, resulting in ZF set theory. The choice axiom was introduced due to the many applications it had, although it resulted from a very simple idea when applied to finite sets. The purpose of Hilbert’s program was to justify all of mathematics by means of a reduction to finitism. It is known that this program failed due to Godel theorems, specially to second Godel theorem that turned difficult the proofs of consistency. Gentzen did one proof of consistency, the consistency of Peano’s arithmetic [7]. But his proof his claimed to use a no finitistic method, transfinite induction up to ǫ0 . We want to present a proof of the consistency of set theory (ZFC). 1
The paper is structured as follows. Section 2 presents set theory. Section 3 presents the incompleteness Godel theorems. Section 4 presents the proof of consistency of set theory. In the last section we present some conclusions and some directions for future work, along with some applications of the result of previous section
2 Set theory We will present the axioms of set theory [4].
Axioms of Zermelo-Fraenkel 1.1 Axiom of Extensionality If X and Y have the same elements, then X = Y . 1.2 Axiom of Pairing For an a and b there exists a set {a, b} that contains exactly a and b. 1.3 Axiom Schema of Separaration If P is a property (with parameter p ), then for any X and p there exists a set Y = {uǫX : P (u, p)} that contains all those uǫX that have property P . S 1.4 Axiom of Union For any X there exists a set Y = X, the union of all elements of X. 1.5 Axiom of Power Set For any X there exists a set Y = P (X), the set of all subsets of X. 1.6 Axiom of Infinity There exists an infinite set. 1.7 Axiom Schema of Replacement If a class F is a function, then for any x there exists a set Y = F (X) = {F (x) : xǫX}. 1.8 Axiom of Regularity Every nonempty set has an ǫ-minimal element. 1.9 Axiom of Choice Every family of nonempty sets has a choice function. The theory with axioms 1.1-1.8 is the Zermelo-Fraenkel axiomatic set theory ZF; ZFC denotes the theory ZF with the Axiom of Choice. 1.10 Axiom Schema of Comprehension (false) If P is a property, then there exists a set Y = {x : P (x)}. This principle, however, is false. 1.11 Russell’s Paradox Consider the set S whose elements are all those (and only those) sets that are not members of themselves: S = {X : X 6 ǫX}.
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Question : Does S belong to S? If S belongs to S then S is not a member of itself, and so S 6 ǫS. On the other hand, if S 6 ǫS, then S belongs to S. In either case, we have contradiction. Thus we must conclude that {X : X 6 ǫX} is not a set, and we must revise the intuitive notion of a set. The safe way to eliminate paaradoxes of this type is to abandon the Schema of Comprehemnsion and keep its weak version, the Schema of Separation: If P is a property, then for any X there exists a set Y = {xǫX : P (x)}. Once we give up the full Comprehension Schema, Russel’s Paradox is no longer a threat; moreover, it provides this useful information: The set of all sets does not exist. (otherwise, apply the Separtation Schema to the property x 6 ǫx.) In other words, it is the concept of the set of all sets that is paradoxical, not the idea of comprehension itself.
3 Goedel Incompleteness Theorems The Godel’s Incompleteness Theorems are very important to modern logic. They are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Goedel in 1931, are important both in mathematical logic and the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert’s program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert’s second problem. The First Incompleteness Theorem shows that no consistent (recursive) extension of Peano Arithmetic is complete: there is a statement which is undecidable in the theory. In particular, if ZFC is consistent, no additional axioms can prove or refute every sentence in the language of set theory. The Second Incompleteness Theorem proves that sufficiently strong mathematical theories such as Peano Arithmetic or ZFC (if consistent) cannot prove its own consistency. Goedel Second Incompleteness implies that it is improvable in ZFC that there is a model of ZFC. In the sequel we shall present some limitations of Goedel’s theorems. The conclusions of Goedel’s theorems are only proven for the formal theories that satisfie the necessary hypothesis. Not all axiom systems satisfy these hypothesis, even when these systems have models that include the natural numbers as a subset. For example, there are first order axiomatizations of Euclidean geometry, of real closed fields, and of arithmetic in which multiplication is not provably total; none of of these meet the hypothesis of Goedel’s theorems. The key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or develop basic properties of the natural numbers. Regarding the third example, many weak systems of arithmetic have been studied which do not satisfy the hypothesis of the second incompleteness theorem, and which are consistent and capable of proving their own consistency.
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Goedel’s theorems only apply to effectively generated (that is, recursively enumerable) theories. If all true statements about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano arithmetic (called true arithmetic) for which none of Goedel’s theorems apply in a meaningful way, because this theory is not recursively enumerable. The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the axioms of those theories themselves. It does not show that the consistency cannot be proved from other axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo-Fraenkel set theory (ZFC), or in theories of arithmetic augmented with transfinite induction, as in Gentzen’s consistency proof.
4 The Proof In the following Theorem we present a proof of the consistency of set theory (ZFC), a first order logic theory. We rely on the fact that NFU set theory is consistent. We know that since 1969 NFU set theory is consistent, as prooved by Jensen [5]. This proof is based on a relative consistency of NFU set theory relative to Peano arithmetic. We refer the following publications [3, 1] on NFU literature. Theorem 4.1 (Consistency of Set Theory) Set Theory is consistent. Proof: Quoting ([3], section 19.2 ”ZFC in the Cantorial Ordinals”, We introduce the usual set theory ZFC (Zermelo-Fraenkel set theory) by showing how to interpret it using the Cantorian ordinals, and stating its axioms as theorems. A Cantorian set is a set according to the set theory invented by Cantor and an ordinal is a particular type of set. Then the concept of Cantorian ordinal used above. Thus in the context of NFU set theory, which is consistent, as presented in [3], we can define a model of set theory ([3], Chapter 19, section 19.2 entitled ”ZFC in the Cantorial Ordinals”): the class of cantorian ordinals, for which the axioms of set theory hold, as proven in theorems refered in the quotation above, in section 19.2. Concluding, if NFU is consistent then set theory is semantically consistent. Set theory is a first order theory, an for this type of theories semantic consistency is equivalent to (syntactic) consistency [2]. So we proved that if NFU is consistent then set theory is consistent, and that set theory is consistent since NFU is consistent. This proof again is based on a relative consistency, this time of set theory relative to NFU set theory. So, we started with consistency of Peano arithmetic, by Gentzen, then consistency of NFU set theory, by Jensen, and of set theory.
5 Conclusions and Future Work Until know people feared what would happen if set theory were inconsistent. In the social network for scientific researchers, called (R)esearch (G)ate (RG), the following 4
question was posed: What happens if mathematics is inconsistent. In the following this question is contextualized (and I quote). By Godel’s incompleteness theorem it is impossible to prove consistency of the current widely accepted foundation of mathematics ZFC within ZFC. But this theorem says nothing about existence or non-existence of a possible formal proof for inconsistency of ZFC within ZFC that means it is possible that some day theorists or other working mathematicians find an inconsistency between two mathematical theorems. My first question is about any possible option which could be chosen by set theorists, logicians and mathematicians in this imaginary situation. Another question is about possible impacts of discovering an inconsistency in mathematics, on philosophy of mathematics and some fields of human knowledge like theoretical physics wich use mathematics extensively. It seems weakening the current axiomatic foundation of mathematics in any sense (including removing a particular axiom or moving to another weaker axiomatic system) causes an expected problem. The apocalyptic view of mathematics if consistency of set theory could not be proved is highlihted. So the proof of consistency of set theory is welcome. In a question posed by me, ”Is the set theory consistent?”, it evolved to a positive answer, which the minimal proof we did corroborates. The proof of consistency of set theory is important since set theory is a known foundation for mathematics. As a consequence, the mathematics are consistent too, i.e. free of contradictions. So, the ultimate desire of Hilbert is achieved. In the future, consistency proofs will benefit from the relativization of second Goedel Theorem and from proofs of relative consistency from theories we want to prove consistent relative to consistent theories.
References [1] A. Enayat. Jensen’s NFU, 40 years later. Oberseminar Mathematiche Logik, Bonn, 2009. [2] V. Halbach. The Logic Manual. Oxford University Press, 2010. [3] M. Randall Holmes. Elementary Set Theory with a Universal Set. BruylantAcademia, 1998. [4] T. Jech. Set theory. The Third Millenium Edition. Springer Monographs in Mathmatics, 2002. [5] R. Jensen. On the consistency of a slight (?) modification of Quine’s New Foundations. Synthese, 19:250–63, 1969. [6] C.J. Rodrigues. The consistency of set theory from the consistency of NFU+infinity+choice set theory using a result of the relative consistency of set theory relative to NBG set theory. Technical Report (R)esearch(G)ate, 2014. [7] A.S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1996.
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