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 .

SQUARES, SCALES AND STATIONARY REFLECTION

2001 ◽  
Vol 01 (01) ◽  
pp. 35-98 ◽  
Author(s):  
JAMES CUMMINGS ◽  
MATTHEW FOREMAN ◽  
MENACHEM MAGIDOR
Keyword(s):  
Continuum Hypothesis ◽  
Pcf Theory ◽  
Consistency Results ◽  
Singular Cardinals ◽  
Combinatorial Principles ◽  
Stationary Set ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic ◽  
The Continuum

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

Continuum hypothesis

2018 ◽  
Author(s):  
Mary Tiles
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Natural Numbers ◽  
Generalized Continuum ◽  
Power Set ◽  
Independence Results ◽  
Infinite Set ◽  
Special Case ◽  
Insight Into ◽  
The Continuum

The ‘continuum hypothesis’ (CH) asserts that there is no set intermediate in cardinality (‘size’) between the set of real numbers (the ‘continuum’) and the set of natural numbers. Since the continuum can be shown to have the same cardinality as the power set (that is, the set of subsets) of the natural numbers, CH is a special case of the ‘generalized continuum hypothesis’ (GCH), which says that for any infinite set, there is no set intermediate in cardinality between it and its power set. Cantor first proposed CH believing it to be true, but, despite persistent efforts, failed to prove it. König proved that the cardinality of the continuum cannot be the sum of denumerably many smaller cardinals, and it has been shown that this is the only restriction the accepted axioms of set theory place on its cardinality. Gödel showed that CH was consistent with these axioms and Cohen that its negation was. Together these results prove the independence of CH from the accepted axioms. Cantor proposed CH in the context of seeking to answer the question ‘What is the identifying nature of continuity?’. These independence results show that, whatever else has been gained from the introduction of transfinite set theory – including greater insight into the import of CH – it has not provided a basis for finally answering this question. This remains the case even when the axioms are supplemented in various plausible ways.

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.

The independence of a weak axiom of choice

1956 ◽  
Vol 21 (4) ◽  
pp. 350-366 ◽  
Author(s):  
Elliott Mendelson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Weak Form ◽  
Axiom Of Choice ◽  
Transfinite Induction ◽  
Exact Nature ◽  
Generalized Continuum ◽  
Power Set ◽  
The Axiom Of Choice ◽  
Weak Axiom

1. The purpose of this paper is to show that, if the axioms of a system G of set theory are consistent, then it is impossible to prove from them the following weak form of the axiom of choice: (H1) For every denumerable set x of disjoint two-element sets, there is a set y, called a choice set for x, which contains exactly one element in common with each element of x. Among the axioms of the system G, we take, with minor modifications, Axioms A, B, C of Gödel [6]. Clearly, the independence of H1 implies that of all stronger propositions, including the general axiom of choice and the generalized continuum hypothesis.The proof depends upon ideas of Fraenkel and Mostowski, and proceeds in the following manner. Let a be a denumerable set of objects Δ0, Δ1, Δ2, …, the exact nature of which will be specified later. Let μj = {Δ2j, Δ2j+1} for each j, c = {μ0, μ1, μ2, …}, and b = [the sum set of a]. By transfinite induction, construct the class Vc which is the closure of b under the power-set operation. For each j, it is possible to define a 1–1 mapping of Vc onto itself with the following properties. The mapping preserves the ε-relation, or, more precisely, .

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