On the existence of an analytic set meeting each compact set in a Borel set

1978 ◽  
Vol 84 (1) ◽  
pp. 5-10 ◽  
Author(s):  
A. R. D. Mathias ◽  
A. J. Ostaszewski ◽  
M. Talagrand
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Continuum Hypothesis ◽  
Compact Set ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Counter Example ◽  
The Axiom Of Choice ◽  
The Continuum

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

Some sets with measurable inverses

1969 ◽  
Vol 65 (2) ◽  
pp. 437-438
Author(s):  
Roy O. Davies
Keyword(s):  
Real Function ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Borel Set ◽  
Analytic Set ◽  
Linear Set ◽  
Measurable Set ◽  
Intermediate Results ◽  
The Axiom Of Choice ◽  
The Continuum

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.

Gödel, Kurt (1906–78)

2018 ◽  
Author(s):  
John W. Dawson
Keyword(s):  
Twentieth Century ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Platonism ◽  
The Axiom Of Choice ◽  
Fundamental Theorems ◽  
The Continuum

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

An addendum to “The work of Kurt Gödel”

1978 ◽  
Vol 43 (3) ◽  
pp. 613-613 ◽  
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 ◽  
The Continuum

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.

The constructible universe

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Constructible Sets ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
The Universe ◽  
The Continuum

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.

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

2010 ◽  
Vol 3 (1) ◽  
pp. 71-92 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Paraconsistent Logic ◽  
Peano Arithmetic ◽  
Large Cardinals ◽  
Cardinal Numbers ◽  
Naive Set Theory ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
The Continuum

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Forcing

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 Continuum

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).

Gödel and Set Theory

2007 ◽  
Vol 13 (2) ◽  
pp. 153-188 ◽  
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 ◽  
The Continuum

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.

Every n-generic degree is a minimal cover of an n-generic degree

1993 ◽  
Vol 58 (1) ◽  
pp. 219-231 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Characteristic Function ◽  
Set Theory ◽  
Initial Segment ◽  
Continuum Hypothesis ◽  
Recursion Theory ◽  
Turing Degree ◽  
Minimal Cover ◽  
The Axiom Of Choice ◽  
Generic Set ◽  
The Continuum

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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