# Questions tagged [gch]

Questions about the generalized continuum hypothesis.

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### Does GCH for alephs imply the axiom of choice?

GCH for alephs means the statement that, for any aleph $\kappa$, there are no cardinals $\mathfrak{r}$ such that $\kappa<\mathfrak{r}<2^\kappa$. Does GCH for alephs imply the axiom of choice? ...
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### How badly can the GCH fail globally?

It's known that we have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms. My question is whether we can have global ...
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### GCH implies acceptability

I have been studying the concept of acceptability, particularly in its relation to GCH. There are many versions of it in the sources I have found, with some slight variations, and some of them are ...
233 views

### Continuum hypothesis in nonstandard universe

In Vladimir Kanovei's book "Nonstandard Analysis, Axiomatically", some nonstandard set theory is introduced. It seems that, one of them, DNST, is useful. When we are talking about higher order ...
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### Question about Jech's proof of V = L implies GCH

On pg 190 of Jech's Set Theory, he proves V = L implies GCH. I understand it all except the following: Thus let X ⊂ $ω_α$. There exists a limit ordinal δ>$ω_α$ such that X ∈ $L_δ$. Let M be an ...
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### Is $V=\textsf{HOD}\not\Rightarrow\textsf{GCH}$ consistent?

Whenever $M$ is some fine-structural $L$-like model we can prove the implication $V=M\Rightarrow\textsf{GCH}$. For $L$ this is due to Gödel, and for the modern extender models it follows simply by ...
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