The generalized continuum hypothesis is equivalent to the generalized maximization principle

1971 ◽  
Vol 36 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Joel I. Friedman
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Methodological Principle ◽  
Theoretical Statement ◽  
Generalized Continuum ◽  
And Mathematics ◽  
Maximization Principle ◽  
The Continuum

In spite of the work of Gödel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis (GCH) from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth (or falsity). Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search for “other (hitherto unknown) axioms of set theory which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts” (see Gödel [7, p. 265]).

Eliminating the continuum hypothesis

1969 ◽  
Vol 34 (2) ◽  
pp. 219-225 ◽  
Author(s):  
Richard A. Platek
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Simple Theory ◽  
Generalized Continuum ◽  
The Continuum

In this paper we show how the assumption of the generalized continuum hypothesis (GCH) can be removed or partially removed from proofs in Zermelo-Frankel set theory (ZF) of statements expressible in the simple theory of types. We assume the reader is familiar with the latter language, especially with the classification of formulas and sentences of that language into Σκη and Πκη form (cf. [1]) and with how that language can be relatively interpreted into the language of ZF.

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.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

Forcing for the impredicative theory of classes

1972 ◽  
Vol 37 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Rolando Chuaqui
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Consistency Proof ◽  
Suitable Form ◽  
Definition Of ◽  
Class Construction ◽  
The Continuum

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

Letter Games: a metamathematical taster

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

2001 ◽  
Vol 66 (4) ◽  
pp. 1766-1782 ◽  
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 ◽  
The Continuum

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().

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