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.

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.

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.

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.

scholarly journals On generalizations of projectivity for modules over Dedekind domains

1981 ◽  
Vol 31 (2) ◽  
pp. 207-216 ◽  
Author(s):  
Jutta Hausen
Keyword(s):  
Exact Sequence ◽  
Structure Theorem ◽  
Continuum Hypothesis ◽  
Projective Modules ◽  
Generating Set ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Dedekind Domains ◽  
The Continuum

AbstractA module M over a ring R is κ-projective, κ a cardinal, if M is projective relative to all exact sequence of R-modules 0 → A → B → C → 0 such that C has a generating set of cardinality less than κ. A structure theorem for κ-projective modules over Dedekind domains is proven, and the κ-projectivity of M is related to properties of ExtR (M, ⊕ R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective и1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced И0-projective Z-module which is not free.

An existence theorem for a special ultrafilter when 𝔡 = 𝔠

1993 ◽  
Vol 58 (4) ◽  
pp. 1359-1364
Author(s):  
James J. Moloney
Keyword(s):  
Lower Bound ◽  
Prime Ideal ◽  
Existence Theorem ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Image Position ◽  
The Continuum ◽  
Convergent Sequences

For an ultrafilter , consider the ultrapower NN/. 〈an〉/ is in the top sky of NN/ if there exists a sequence 〈bn〉 ∈ NN such thatandIn [M2] we showed, assuming the Continuum Hypothesis, that there are exactly 10 c/p's (where c is the ring of real convergent sequences and p is a prime ideal of c). To get the lower bound we showed that there will be at least 10 c/p's in any model of ZFC where there exist both of the following kinds of ultrafilter:(i) nonprincipal P-points,(ii) non-P-points such that when the top sky is removed from NN/, the remaining model has countable cofinality.In [M2] we showed that the Continuum Hypothesis implies the existence of the ultrafilter in (ii). In this paper we show that its existence is implied by an axiom weaker than the Continuum Hypothesis, in fact weaker than Martin's Axiom, namely,(*) If is a subset of NN such that for any f: N → N there exists g ∈ such that g(n) > f(n) for all n, then ∣∣ = .

Remarks on Martin's Axiom and the Continuum Hypothesis

1991 ◽  
Vol 43 (4) ◽  
pp. 832-851 ◽  
Author(s):  
Stevo Todorcevic
Keyword(s):  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Image Position ◽  
The Continuum

Martin's axiom and the Continuum Hypothesis are studied here using the notion of accc partitioni.e., a partition of the formwhereK0has the following properties:(a)K0contains subsets of its elements as well as all singletons ofX.(b) Every uncountable subset of K0contains two elements whose union is inK0.

scholarly journals Some Consequences of Martin’s Axiom and the Negation of the Continuum Hypothesis

1973 ◽  
Vol 49 ◽  
pp. 117-125 ◽  
Author(s):  
Juichi Shinoda
Keyword(s):  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

W. Sierpisnki [3] demonstrated 82 propositions, called C1-C82, with the aid of the continuum hypothesis. D. A. Martin and R. M. Solovay remarked in [2] that 48 of these propositions followed from Martin’s axiom (MA), 23 were refuted by and three were independent of But the relation of the remaining eight propositions to has been unsettled.

Countable partitions of Euclidean space

1996 ◽  
Vol 120 (1) ◽  
pp. 07-12 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Euclidean Space ◽  
Continuum Hypothesis ◽  
Isosceles Triangle ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

Erdös has asked whether the plane ℝ2, or more generally n-dimensional Euclidean space ℝn, can be partitioned into countably many sets none of which contains the vertices of an isosceles triangle. Assuming the Continuum Hypothesis (CH), Davies[2] (for n = 2) and Kunen[10] (for arbitrary n) proved that such partitions exist. Assuming Martin's Axiom, Erdös and Komjáth proved in [5] that such partitions exist for n = 2. We will prove here, without additional set-theoretic hypotheses, that there are such partitions in all dimensions.Let ‖x‖ denote the usual Euclidean norm of a point x ∈ ℝn, so that ‖x − y‖ is the distance between x and y.

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).

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