Sheaf theory and the continuum hypothesis

1972 ◽  
pp. 13-42 ◽  
Author(s):  
Myles Tierney
Keyword(s):  
Continuum Hypothesis ◽  
Sheaf Theory ◽  
The Continuum

Flow and Solute Transport in Saturated Porous Media: 1. The Continuum Hypothesis

2008 ◽  
Vol 11 (4) ◽  
pp. 403-413 ◽  
Author(s):  
Amgad Salama ◽  
Paul J. Van Geel
Keyword(s):  
Porous Media ◽  
Solute Transport ◽  
Continuum Hypothesis ◽  
Saturated Porous Media ◽  
The Continuum

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 .

Kreisel, the continuum hypothesis and second order set theory

1976 ◽  
Vol 5 (2) ◽  
Author(s):  
Thomas Weston
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Second Order ◽  
The Continuum ◽  
Order Set

The Operational Significance of the Continuum Hypothesis in the Theory of Water Movement Through Soils and Aquifers

1984 ◽  
Vol 20 (5) ◽  
pp. 521-530 ◽  
Author(s):  
Philippe Baveye ◽  
Garrison Sposito
Keyword(s):  
Water Movement ◽  
Continuum Hypothesis ◽  
The Continuum

Regular Conditional Expectations and the Continuum Hypothesis

1994 ◽  
pp. 85-93
Author(s):  
I. V. Evstigneev
Keyword(s):  
Continuum Hypothesis ◽  
Conditional Expectations ◽  
The Continuum

HAS THE CONTINUUM HYPOTHESIS BEEN SETTLED?

2006 ◽  
pp. 56-75
Author(s):  
MATTHEW FOREMAN
Keyword(s):  
Continuum Hypothesis ◽  
The Continuum

Some remarks on the partition calculus of ordinals

1999 ◽  
Vol 64 (2) ◽  
pp. 436-442 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Regular Cardinal ◽  
Continuum Hypothesis ◽  
Alternative Proof ◽  
Partition Relation ◽  
Negative Relation ◽  
Partition Calculus ◽  
Counter Example ◽  
The Continuum

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

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