# Questions tagged [gch]

Questions about the generalized continuum hypothesis.

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### 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 ...

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### 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, ...

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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 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 ...

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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 ...

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### 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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### Does $V = \textit{Ultimate }L$ imply GCH?

In his Midrasha Mathematicae lectures ("In Search of Ultimate $L$", BSL 23 [2017]: 1–109), Woodin notes that $V = \textit{Ultimate }L$ implies $\textrm{CH}$ (Theorem 7.26, p.103). Is it known whether $...

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### $GCH$ and special Aronszajn trees

Question. Does $\text{GCH}$ imply the existence of a non-special $\aleph_2$-Aronszajn tree ?
Remark 1. By a result of Jensen, it is consistent that $\text{GCH}$ holds and all $\aleph_1$-Aronszajn ...

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### Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa &...

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### Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? [duplicate]

The following statement cannot be proven in $\mathsf{ZFC}$:
(S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$.
Obviously, $\mathsf{...

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### Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists?

In Chang and Keisler's Model Theory I came across the following theorem (Theorem 7.2.13):
Theorem There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, $\sigma$...

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### Failure of GCH at indescribable cardinals

Can $\Pi^m_n$ indescribable cardinal be the first one where $\text{GCH}$ fails?
Hauser showed in
Hauser,K.: Indescribable cardinals and elementary embeddings.
J. Symb. Logic 56, 439457 (1991)
that ...

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### do behavior of gimel or GCH determine all infinte products of cardinals?

Let $Card$ be the class of infinite cardinals and $p\colon Card^2\to Card$ be given by $(\kappa,\lambda)\mapsto\kappa^\lambda$.
Assuming GCH it is known that $p(\kappa,\lambda)$ is either $\kappa$ (if ...

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### The canonical forcing of the GCH and direct limits.

The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds.
The ...

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### Failure of the GCH

What is the (currently known) consistency strength of global failure of the GCH?
I do not have access to the exact statement of the original Foreman-Woodin result. My searches seem to indicate that ...

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### When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)

Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these ...

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### What is the general opinion on the Generalized Continuum Hypothesis?

I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this.
From what I've seen, model theorists and logicians ...