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.

Martin's axiom and Hausdorif measures

1974 ◽  
Vol 75 (2) ◽  
pp. 193-197 ◽  
Author(s):  
A. J. Ostaszewski
Keyword(s):  
Metric Space ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
The Continuum

AbstractA theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

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).

Uncountable Discrete Sets in Extensions and Metrizability

1982 ◽  
Vol 25 (4) ◽  
pp. 472-477 ◽  
Author(s):  
Murray Bell ◽  
John Ginsburg
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuum Hypothesis ◽  
Vietoris Topology ◽  
Discrete Subset ◽  
Similar Argument ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

AbstractIf X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

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().

Some points in βN

1976 ◽  
Vol 80 (3) ◽  
pp. 385-398 ◽  
Author(s):  
Kenneth Kunen
Keyword(s):  
Limit Point ◽  
Continuum Hypothesis ◽  
Stone Space ◽  
Measure Algebra ◽  
Random Real ◽  
Martin’S Axiom ◽  
Real Model ◽  
Limit Points ◽  
Set Of Points ◽  
The Continuum

AbstractAssuming either the continuum hypothesis or Martin's axiom, we show that in the space βN – N, there are: (a) points which are not P-points, but which are also not limit points of any countable set, and (b) a countable set of points dense in itself such that each of the points is not a limit point of any countable discrete set. Our method is to construct such points in the Stone space of a measure algebra, and then embed that Stone space into βN – N. We also, by a similar use of measures, establish the independence of the existence of selective ultrafilters by showing that there are none in the random real model.

The Roles of Set Theories in Mathematics

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.

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

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