Gödel's First Proof of the Consistency of the Axiom of Choice

The so called Generalized Continuum Hypothesis (GCH) is the sentence: "If A is an infinile set whose cardinal number is K and 2K denotes the cardinal number of the set P(A) of subsets of A (the power set of A), and K + denotes the succesor cardinal of K, then 2K = K +". The Continuum Hypothesis (CH) asserts the particular case K = o. It is clear that GCH implies CH. Another equivalent version of GCH, is the sentence: 'Any subset of the set of subsets of a given infinite set is or of cardinality less or equal than the cardinality of the given set, or of the cardinality of all the set of subsets". Gödel in 1939, and Cohen in 1963, settled the relative consistency of the Axiom of Choice (AC) and of its negation not-AC, respectively, with respecllo the Zermelo-Fraenkel set theory (ZF). On the other hand, Gödel in 1939, and Cohen in 1963 settled too, the relative consistency of GCH , CH and of its negations not-GCH, not-CH, respectively, with respect to the Zermelo-Fraenkel set theory with the Axiom of Choice (ZF + AC or ZFC). From these results we know that GCH and AC are undecidable sentences in ZF set theory and indeed, the most famous undecidable sentences in ZF; but, which is the relation between them? From the above results, in the theory ZF + AC is not demonstrated GCH; it is clear then that AC doesn't imply GCH in ZF theory, Bul does GCH implies AC in ZF theory? The answer is yes! or equivalently, there is no model of ZF +GCH + not-AC. A very easy proof can be given if we have an adecuate definition of cardinal number of a set, that doesn't depend of AC but depending from the Regularity Axiom, which asserls that aIl sets have a range, which is an ordinal number associated with its constructive complexity. We define the cardinal number of A, denoted |A|, as foIlows: |A|= { The least ordinal number equipotent with A, if A is well orderable The set of all sets equipotent with A and of minimum range, in other case. It is clear that without AC, may be not ordinal cardinals and all cardinals are ordinal cardinals if all sets are well orderable (AC). Now we formulate: GCH*: For all ordinal cardinal I<, 2K = I< + In the paper is demonstrated that this formulation GCH* is implied by the traditional one, and indeed equivalent to it. Lemma, The power set of any well orderable set is well orderable if and only if AC. This is one of the many equivalents of AC in ZF,due lo Rubin, 1960. Proposition. In ZF is a theorem: GCH* implies AC. Supose GCH*. Let A be a well orderable set; then |A| = K an ordinal cardinal, so A is equipotent with K and then P~A) is equipotent with P(K); therefore |P(A)I|= |P(K)| = 2K = K+. But then |P(A)|= K+ and P(A) 'is equipotent with K+ and K+ is an ordinal cardinal; therefore P(A) is well orderable with the well order induced by means of the bijection, from the well order of K+. Corolary: In ZF are theorems: GCH impIies AC and GCH is equivalent to GCH*. We see from this proof, that GCH asserts that the cardinal number of the power set of a well orderable set A is an ordinal, which is equivalent to AC, but GCH asserts also that that ordinal cardinal is |A|+ , the ordinal cardinal succesor of the ordinal cardinal of the well orderable set A.

A Proof of the Relative Consistency of the Continuum Hypothesis

Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

The Continuum Hypothesis, originally posited by the pioneering mathematician Georg Cantor in the latter part of the 19th century, stands as a cornerstone inquiry in the realm of set theory. This paper embarks on a journey, delving into the rudiments of set theory, before tracing the evolutionary trajectory of the Continuum Hypothesis. Central to this exploration is the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) a foundational pillar in modern set theoretic studies. The core tenets of ZFC are dissected, shedding light on the seminal proofs presented by luminaries in the field that underline the unprovability of the CH within this axiomatic system. Beyond its mathematical intricacies, the paper underscores the profound philosophical and practical implications of the CH in both set theory and the broader mathematical landscape. In synthesizing these insights, a profound realization emerges: the inherent limitations in establishing the veracity of the Continuum Hypothesis within the confines of ZFC. This poignant revelation beckons deeper introspection into the foundational underpinnings of mathematics, stirring both intrigue and reflection amongst scholars and enthusiasts alike.

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

The effectivity of existential statements in axiomatic set theory

We believe that it is possible to put the whole work of Bourbaki into a computer. One of the objectives of the Gaia project concerns homological algebra (theory as well as algorithms); in a first step we want to implement all nine chapters of the book Algebra. But this requires a theory of sets (with axiom of choice etc.) more powerful than what is provided by Ensembles; we have chosen the work of Carlos Simpson as basis. This reports lists and comments all definitions and theorems of the Chapter ''Theory of Sets''. The code (including almost all exercises) is available on the Web, under http://www-sop.inria.fr/marelle/gaia . Version one was released in July 2009, version 2 in December 2009, version 3 in March 2010. Version 4 is based on the Coq ssreflect library. In version 5, released in December 2011, the ''iff_eq'' axiom has been withdraw, and the axiom of choice modified. Version 6 was released in October 2013. Version 7 was released in December 2018

INDEPENDENCE RESULTS IN SET THEORY

The distinguished mathematician Saunders Mac Lane has titled his talk at these meetings Algebra as a means of understanding mathematics. Had I known about his title, I would have called this talk Set theory as a means of doing mathematics. These titles represent quite a historical change. Speaking simplistically, and exaggerating somewhat, seventy-five years ago algebra was something mathematicians did largely for its own sake, while set theory belonged to that quasi-philosophical branch of mathematics known as foundations, whose main purpose seemed to be to convince the phi losophical ly inclined that mathemat ic ians should be allowed to continue doing what they had always done, and which most mathematicians felt they could safely ignore. As with all historical generalizations, this is, of course, not entirely accurate. I will let the algebraists speak for algebra (such matters as groups and crystallography come immedia te ly to mind). As for set theory, its roots lie much more in mathematics than in philosophy. Cantor invented ordinals while trying to solve a question about Fourier series on which mathematicians are actively working today--Kechr is and Louveau have an excellent book on the subject. Early on, Polish and Russian mathematicians invented descriptive set theory, the study of the structure of the reals in set-theoretic fashion. And point-set topology, since its inception, has been inextricably intertwined with set theory. Even Hilbert's problems did not avoid all reference to set theory. But it remained true that most mathematicians could get by quite well with union, intersection, and (if necessary) the axiom of choice, learned as quickly as possible, preferably in the preface to a book about some other area of mathematics. The thesis of this talk is that this situation has changed drastically. Hard-core, mainstream mathematics (defined as: the mathematics most mathematicians think doesn't use set theory) is beginning to see results using modern set theoretic techniques. The purpose of this talk is to introduce you to a few of

This paper is an informal exposition of two significant results in the foundations of set theory: G?del's 1938 proof that the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) are consistent with the axioms of Zermelo-Fraenkel set theory (ZF), and Cohen's 1963 proof of the consistency of the denial of GCH and AC. These theorems amount jointly to the independence of GCH and AC. It is our conviction that these results, and the novel methods they employ, occupy a position like that of the more celebrated incompleteness results for arithmetic. Adequately informed discussion of a broad range of issues in foundations of mathematics and epistemology generally is impossible without some idea of these independence proofs. Moreover, this work sheds retrospective light on much of the metamathematical work that preceded it. Philosophers are therefore obligated to acquire some knowledge of it. While these results are not 'recent' by the standards of a fast-moving discipline like logic, the formidable technicalities involved have kept them inaccessible to non-logicians. We hope to improve the situation. Important details have been relegated to an Appendix, to whose nth entry 'An' refers. Only A16 is strictly necessary for a full grasp of the text.

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics.
 Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic.
 In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known.
 As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.

Gödel [3] published a monograph in 1940 proving a highly significant theorem, namely that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert’s famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5,6]. Obviously, the theorem’s significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. Gödel [3, p. 33] states it as follows, using Σ to denote the axioms for set theory:What we shall prove is that, if a contradiction from the axiom of choice and the generalized continuum hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σ alone.Gödel presents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary

Categorical set theory: A characterization of the category of sets

The forcing technique was discovered by Paul Cohen in the early sixties. Since then forcing has appeared to be a very powerful tool to provide independence results in Set theory. Actually, because of the foundational role played by Set theory with regard to the rest of classical mathematics, and because of the possibility to mimic from the standard axiomatic basis of Set theory, ZFC, the proof of the existence of almost any mathematical object, forcing has been applied to different areas of mathematics revealing to us the undecidability of many different important questions connected with different branches of mathematics. Given the pervasive presence of the independence phenomenon in Set theory determined by forcing, a natural philosophical question arises: is forcing the ultimate horizon of Set theory, or is it (as a source of undecidability) to be considered as a pathology that needs to be neutralised? A special kind of results in Set theory, known in literature as generic absoluteness results, give mathematical substance to the perspective that the real challenge that the discovery of the forcing technique places to the set theorist, as well as to the philosopher of mathematics, goes beyond the idea that the right answer to questions such as the Continuum Hypothesis is given by computing the precise extent of their undecidability. In fact, when it is possible to relieve generic absoluteness for a certain mathematical structure, a different framework appears where forcing can be exploited and, so we may say, integrated into the practice of the mathematician as a strong tool for proving theorems. In chapter 1 of my dissertation I recall some main aspects of the forcing technique developed following the Boolean valued-models approach introduced by Scott, Solovay, and Vopenka starting from 1965. In chapters 2 and 3 I analyze some main motivations behind Viale's and Woodin's alternative strategies for producing generic absoluteness for the structure at the level of the Continuum problem. I try to stress, in particular, the essential use of the so-called forcing axioms that is inherent Viale's generic absoluteness results and that, to some extent, conflicts with Woodin's choice to introduce a strong logic as the appropriate setting for studying the possibility of generic absoluteness at the level of the Continuum problem. In chapter 4 I open the philosophical discussion and I try to correlate the pure mathematical phenomenon of generic absoluteness described in chapters 2 and 3 to the more general philosophical debate concerning the question of Pluralism in Set theory and the search for new axioms. Insofar as we are interested in spell out Viale's and Woodin's absoluteness results in terms of the right axiomatisation for the structure theory where the Continuum problem is expressible, I try to sketch an argument according to which the possibility to unify the two distinct theories offered by Viale and Woodin emerges as one of notable philosophical importance.