# Tagged Questions

Questions about the generalized continuum hypothesis.

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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 ...
### 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 ...