The constructible universe

the ‘universe’ of constructible sets was introduced by Kurt Gödel in order to prove the consistency of the axiom of choice (AC) and the continuum hypothesis (CH) with the basic (ZF) axioms of set theory. The hypothesis that all sets are constructible is the axiom of constructibility (V = L). Gödel showed that if ZF is consistent, then ZF + V = L is consistent, and that AC and CH are provable in ZF + V = L.

An addendum to “The work of Kurt Gödel”

10.2307/2273536 ◽

1978 ◽

Vol 43 (3) ◽

pp. 613-613 ◽

Cited By ~ 3

Author(s):

Stephen C. Kleene

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Axiom System ◽

Von Neumann ◽

Main Achievement ◽

Constructible Sets ◽

Kurt Gödel ◽

The One ◽

The Axiom Of Choice ◽

Gödel has called to my attention that p. 773 is misleading in regard to the discovery of the finite axiomatization and its place in his proof of the consistency of GCH. For the version in [1940], as he says on p. 1, “The system Σ of axioms for set theory which we adopt [a finite one] … is essentially due to P. Bernays …”. However, it is not at all necessary to use a finite axiom system. Gödel considers the more suggestive proof to be the one in [1939], which uses infinitely many axioms.His main achievement regarding the consistency of GCH, he says, really is that he first introduced the concept of constructible sets into set theory defining it as in [1939], proved that the axioms of set theory (including the axiom of choice) hold for it, and conjectured that the continuum hypothesis also will hold. He told these things to von Neumann during his stay at Princeton in 1935. The discovery of the proof of this conjecture On the basis of his definition is not too difficult. Gödel gave the proof (also for GCH) not until three years later because he had fallen ill in the meantime. This proof was using a submodel of the constructible sets in the lowest case countable, similar to the one commonly given today.

Gödel and Set Theory

Bulletin of Symbolic Logic ◽

10.2178/bsl/1185803804 ◽

2007 ◽

Vol 13 (2) ◽

pp. 153-188 ◽

Cited By ~ 10

Author(s):

Akihiro Kanamori

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Order Logic ◽

Point Of View ◽

First Order ◽

Kurt Gödel ◽

The Axiom Of Choice ◽

Universe Of Sets ◽

The Universe ◽

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.

Kurt Gödel. The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. (Lectures delivered at the Institute for Advanced Study 1938–1939; notes by George W. Brown.) Annals of Mathematics studies, no. 3. Lithoprinted. Princeton University Press, Princeton1940, 66 pp.

10.2307/2268607 ◽

1941 ◽

Vol 6 (3) ◽

pp. 112-114 ◽

Cited By ~ 1

Author(s):

Paul Bernays

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Axiom Of Choice ◽

Princeton University ◽

Advanced Study ◽

Kurt Gödel ◽

Generalized Continuum ◽

The Axiom Of Choice

Kurt Gödel. The consistency of the axiom of choice and the generalized continuum-hypothesis with the axioms of set theory. Annals of Mathematics studies, no. 3. Second printing, lithoprinted. Princeton University Press, Princeton1951, 69 pp.

10.2307/2267705 ◽

1952 ◽

Vol 17 (3) ◽

pp. 207-208

Author(s):

Leon Henkin

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Axiom Of Choice ◽

Princeton University ◽

Kurt Gödel ◽

Generalized Continuum ◽

The Axiom Of Choice

Forcing

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y078-1 ◽

2018 ◽

Author(s):

John P. Burgess

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Axiom Of Choice ◽

Forcing Axioms ◽

Martin's Axiom ◽

Martin’S Axiom ◽

The Axiom Of Choice ◽

The method of forcing was introduced by Paul J. Cohen in order to prove the independence of the axiom of choice (AC) from the basic (ZF) axioms of set theory, and of the continuum hypothesis (CH) from the accepted axioms (ZFC = ZF + AC) of set theory (see set theory, axiom of choice, continuum hypothesis). Given a model M of ZF and a certain P∈M, it produces a ‘generic’ G⊆P and a model N of ZF with M⊆N and G∈N. By suitably choosing P, N can be ‘forced’ to be or not be a model of various hypotheses, which are thus shown to be consistent with or independent of the axioms. This method of proving undecidability has been very widely applied. The method has also motivated the proposal of new so-called forcing axioms to decide what is otherwise undecidable, the most important being that called Martin’s axiom (MA).

Extended ultrapowers and the Vopěnka-Hrbáček theorem without choice

10.2307/2274701 ◽

1991 ◽

Vol 56 (2) ◽

pp. 592-607 ◽

Cited By ~ 2

Author(s):

Mitchell Spector

Keyword(s):

Set Theory ◽

Fundamental Theorem ◽

Axiom Of Choice ◽

Strongly Compact Cardinal ◽

Compact Cardinal ◽

Axiom Of Determinacy ◽

Omitting Types ◽

The Axiom Of Choice ◽

The Universe ◽

AbstractWe generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof of Vopěnka and Hrbáček used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in L[2ω], there cannot exist a θ-compact cardinal less than θ (where θ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form L[A]. The result for L[2ω] is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem.

Constructible Sets with Applications - Studies in Logic and the Foundations of Mathematics ◽

Chapter VII Consistency of the Axiom of Choice and of the Continuum Hypothesis

10.1016/s0049-237x(09)70058-8 ◽

1969 ◽

pp. 99-119

Keyword(s):

Continuum Hypothesis ◽

Axiom Of Choice ◽

The Axiom Of Choice ◽

On Arbitrary sets andZFC

Bulletin of Symbolic Logic ◽

10.2178/bsl/1309952318 ◽

2011 ◽

Vol 17 (3) ◽

pp. 361-393 ◽

Cited By ~ 8

Author(s):

José Ferreirós

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Real Numbers ◽

The Real ◽

Formal Systems ◽

Infinite Sets ◽

The Axiom Of Choice ◽

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

Letter Games: a metamathematical taster

The Mathematical Gazette ◽

10.1017/mag.2016.109 ◽

2016 ◽

Vol 100 (549) ◽

pp. 442-449

Author(s):

A. C. Paseau

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Incompleteness Theorem ◽

Mathematical Study ◽

Mathematical System ◽

Kurt Gödel ◽

Standard Set ◽

The Continuum ◽

Simplified Form

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

Power-like models of set theory

10.2307/2694973 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1766-1782 ◽

Cited By ~ 3

Author(s):

Ali Enayat

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Mahlo Cardinal ◽

Uncountable Cardinal ◽

Consistent Extension ◽

Elementary Submodel ◽

Finite Language ◽

The Universe ◽

Models Of Set Theory ◽

Abstract.A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ1-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2<0 = 0 then every consistent extension of ZF also has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ2-like model. Here we prove:Theorem A. If 0 has the tree property then the following are equivalent for any completion T of ZFC:(i) T has a 0-like model.(ii) Ф ⊆ T. where Ф is the recursive set of axioms {∃κ (κ is n-Mahlo and “Vκis a Σn-elementary submodel of the universe”): n ∈ ω}.(iii) T has a λ-like model for every uncountable cardinal λ.Theorem B. The following are equiconsistent over ZFC:(i) “There exists an ω-Mahlo cardinal”.(ii) “For every finite language , all ℵ2-like models of ZFC() satisfy the schemeФ().