THE COUNTERPARTS TO STATEMENTS THAT ARE EQUIVALENT TO THE CONTINUUM HYPOTHESIS

2015 ◽  
Vol 80 (4) ◽  
pp. 1075-1090
Author(s):  
ASGER TÖRNQUIST ◽  
WILLIAM WEISS
Keyword(s):  
Continuum Hypothesis ◽  
Partition Relations ◽  
Image Position ◽  
The Continuum

AbstractWe consider natural ${\rm{\Sigma }}_2^1$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these ${\rm{\Sigma }}_2^1$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for ${\rm{\Sigma }}_2^1$ colourings which hold precisely when there is a non-constructible real.

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.

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.

An Elementary Approach to the Fine Structure of L

1997 ◽  
Vol 3 (4) ◽  
pp. 453-468 ◽  
Author(s):  
Sy D. Friedman ◽  
Peter Koepke
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Central Place ◽  
Structure Theory ◽  
Original Form ◽  
Elementary Embedding ◽  
Elementary Model ◽  
Image Position ◽  
The Continuum

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some , < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L.

Martin's axiom and the continuum

1995 ◽  
Vol 60 (2) ◽  
pp. 374-391 ◽  
Author(s):  
Haim Judah ◽  
Andrzej Rosłanowski
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Point Of View ◽  
Georg Cantor ◽  
Mathematical Research ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Infinite Set ◽  
The Continuum

Since Georg Cantor discovered set theory the main problem in this area of mathematical research has been to discover what is the size of the continuum. The continuum hypothesis (CH) says that every infinite set of reals either has the same cardinality as the set of all reals or has the cardinality of the set of natural numbers, namelyIn 1939 Kurt Gödel discovered the Constructible Universe and proved that CH holds in it. In the early sixties Paul Cohen proved that every universe of set theory can be extended to a bigger universe of set theory where CH fails. Moreover, given any reasonable cardinal κ, it is possible to build a model where the continuum size is κ. The new technique discovered by Cohen is called forcing and is being used successfully in other branches of mathematics (analysis, algebra, graph theory, etc.).In the light of these two stupendous works the experts (especially the platonists) were forced to conclude that from the point of view of the classical axiomatization of set theory (called ZFC) it is impossible to give any answer to the continuum size problem: everything is possible!In private communications Gödel suggested that the continuum size from a platonistic point of view should be ω2, the second uncountable cardinal. As this is not provable in ZFC, Gödel suggested that a new axiom should be added to ZFC to decide that the cardinality of the continuum is ω2.

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 .

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