# Questions tagged [gch]

Questions about the generalized continuum hypothesis.

Questions about the generalized continuum hypothesis.

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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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0
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GCH can fail anywere in HOD
Can exist model $M$ where for any Easton's function $\gamma=f(\kappa)$ exist $X$ what $M\vDash HOD(X)\vDash "2^\kappa=\gamma"$ ?
Even for singular cardinals?

10
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1
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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?
...

22
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1
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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 ...

5
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1
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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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0
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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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2
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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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1
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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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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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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 ...

11
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1
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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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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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1
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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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1
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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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4
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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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1
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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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4
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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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4
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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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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 ...