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Results tagged with continuum-hypothesis
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user 95347
Questions about the continuum hypothesis, or where the continuum hypothesis or its negation plays a role. This tag is also suitable, by extension, to refer to the generalized continuum hypothesis and related issues.
11
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2
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Can GCH fail everywhere every way?
The following question is about if it is compatible to add to $\sf ZF$ an axiom asserting the existence of a countable transitive model of $\sf ZF$ such that for every strictly increasing function $f$ …
- 11.3k
4
votes
1
answer
527
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How to settle the Generalized Continuum Hypothesis when there are urelements?
Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I …
- 11.3k
2
votes
1
answer
178
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If GCH is breached the same way before a singular of uncountable cofinality, would that brea...
By 1 step breach of the GCH I mean the following: $$ 2^{\aleph_{\alpha}} = \aleph_{\alpha+2}$$
Now, it is known that there are more constrains on the cardinality of power sets at singlular cardinals t …
- 11.3k
2
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0
answers
159
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What is the consistency strength of the following pattern of failure of the continuum hypoth...
What is the least theory in which the following sentence is proved?
$ \exists M: M\text { is CTM(ZFC+ GCH)} \land \forall \kappa \in Card^M (\kappa > 1 \implies \\\exists N: N \text { is CTM(ZFC) } …
- 11.3k
1
vote
1
answer
148
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Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of...
In an earlier positing to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in every way is to the negative, this is the case in $\sf ZFC$, however it also appears that …
- 11.3k
1
vote
0
answers
182
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Can GCH fail everywhere in every finite way?
Since the $\sf GCH$ cannot fail everywhere everyway (see here), the question here is if it can fail everywhere in every finite manner, that if we have a strictly increasing function $f$ on the ordinal …
- 11.3k
1
vote
1
answer
181
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What's the consistency status/strength of this limitation principle?
$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, …
- 11.3k
0
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0
answers
136
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What's the consistency strength of resemblance + global failure of the continuum hypothesis?
Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:
Extensionality: $\forall z (z \in x \left …
- 11.3k
-2
votes
1
answer
173
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Which extension of ZFC proves that ZFC can only prove CH satisfied by the first two sets?
Which extension of $\sf ZFC$ prove that
$$ {\sf ZFC} \not \vdash \exists x \, ( \operatorname {CH}(x) \land x \neq \emptyset \land x \neq 1)$$
Where $\operatorname {CH}(x) \iff \neg \exists \kappa \, …
- 11.3k