Kreisel, the continuum hypothesis and second order set theory

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

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Set theoretic properties of Loeb measure

Journal of Symbolic Logic ◽

10.2307/2274471 ◽

1990 ◽

Vol 55 (3) ◽

pp. 1022-1036 ◽

Cited By ~ 2

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.

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Forcing for the impredicative theory of classes

Journal of Symbolic Logic ◽

10.2307/2272540 ◽

1972 ◽

Vol 37 (1) ◽

pp. 1-18 ◽

Cited By ~ 9

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 .

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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’.

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Power-like models of set theory

Journal of Symbolic Logic ◽

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 ◽

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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The Roles of Set Theories in Mathematics

10.1093/oso/9780198748991.003.0001 ◽

2018 ◽

Author(s):

Colin McLarty

Keyword(s):

Set Theory ◽

Gauge Theories ◽

Elementary Theory ◽

Continuum Hypothesis ◽

Large Cardinals ◽

The Continuum

What mathematicians know and use about sets varies across branches of mathematics but rarely includes such fundamental aspects of Zermelo–Fraenkel (ZF) set theory as the iterative hierarchy. All mathematicians know and use the axioms of the Elementary Theory of the Category of Sets (ETCS), though few know ETCS or any set theory by name. The chapter depicts the iterative hierarchy of ZF and constructibility as gauge theories. Since gauge theories are prominently used in physics, so these are used in work on the continuum hypothesis, large cardinals, and provability in arithmetic. But mathematicians outside logic avoid these gauges and work with structures only up to isomorphism, as does ETCS.

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MINIMUM MODELS OF SECOND-ORDER SET THEORIES

Journal of Symbolic Logic ◽

10.1017/jsl.2019.27 ◽

2019 ◽

Vol 84 (02) ◽

pp. 589-620

Author(s):

KAMERYN J. WILLIAMS

Keyword(s):

Set Theory ◽

Second Order ◽

Countable Model ◽

Main Question ◽

Minimal Models ◽

Transitive Model ◽

Transfinite Recursion ◽

Order Set ◽

Countable Models

AbstractIn this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.

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Gödel, Kurt (1906–78)

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y084-1 ◽

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.

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Set Theory and C*-Algebras

Bulletin of Symbolic Logic ◽

10.2178/bsl/1174668215 ◽

2007 ◽

Vol 13 (1) ◽

pp. 1-20 ◽

Cited By ~ 10

Author(s):

Nik Weaver

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

C Algebra ◽

Calkin Algebra ◽

C Algebras ◽

Basic Object ◽

The Continuum

AbstractWe survey the use of extra-set-theoretic hypotheses, mainly the continuum hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic object of interest.

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