# Questions tagged [gch]

Questions about the generalized continuum hypothesis.

19 questions
Filter by
Sorted by
Tagged with
2k views

### Is GCH useful in proving theorems?

By GCH I mean the Generalized Continuum Hypothesis. Let me give some context before presenting my question. When the axiom of choice was introduced by Zermelo in his 1904 proof of Well-Ordering ...
• 2,673
454 views

### What is known about the consistency of $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$ for all $\alpha$?

For $\gamma$ an ordinal, let “$H_\gamma$” be the statement: For all ordinals $\alpha$, we have $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$. So clearly $H_0$ is false, and so is $H_\omega$; in fact, ...
• 29.7k
636 views

### 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? ...
• 1,288
888 views

### How badly can the GCH fail globally?

It's known that we can 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 ...
• 1,208
383 views

### 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 ...
• 2,317
258 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 ...
• 475
373 views

### 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 ...
335 views

### 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 ...
• 1,162
2k views

• 18k