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I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:

Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?

Is the presumably affirmative answer possible without any semi-axioms like CH or measurable cardinals?

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    $\begingroup$ I believe this question is coming from a non-expert in set theory looking for advice from people who are experts. $\endgroup$
    – David Roberts
    Feb 18 at 10:56

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The possibility of changing the cofinality of a regular cardinal without collapsing any cardinals is equiconsistent with a measurable cardinal:

On one hand, if $\kappa$ is measurable, then by Prikry Forcing one obtains a model in which $\kappa$ is singular of cofinality $\omega$ and all cardinals are preserved.

On the other hand, work by Dodd and Jensen shows that if $\kappa$ is regular but becomes singular in some generic extension where all cardinals are preserved, there is an inner model where $\kappa$ is measurable.

EDIT: $K$ is the so-called core model which is an inner model similar to the constructible universe $L$. Notably, $K$ is not changed by set forcing (this has been stated in Consistency strength of some problems about singular cardinals which answers the same question but i do not know an exact reference). So suppose $\kappa$ is regular in the model $V$ and singular in some set forcing extension $V[G]$. Then in particular $\kappa$ is regular in $K^{V[G]}$ (the core model constructed in $V[G]$) which equals $K^V$ by the preceding remark. As $K^V\subseteq V$, $\kappa$ is regular in $K$, hence measurable.

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  • $\begingroup$ Some references to literature would be welcome. $\endgroup$ Feb 18 at 15:25
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    $\begingroup$ The first fact is shown e.g. in Jechs book on set theory (Theorem 21.10). The second fact is much harder to show and is proven e.g. in Mitchell's Chapter "The Covering Lemma" in the Handbook of Set Theory (Theorem 2.5). $\endgroup$ Feb 18 at 15:44
  • $\begingroup$ The Theorem 2.5 on the page 1506 in HST is a bit technical and it is not apparent how from it follows that if $\kappa$ is regular but becomes singular in some generic extension where all cardinals are preserved, there is an inner model where $\kappa$ is measurable. Could you write (perhaps as an update to your answer) a step by step derivation on how it follows form thm 2.5 your claim above in this comment ? And perhaps also how equiconsistency is derived ? $\endgroup$ Feb 20 at 12:25
  • $\begingroup$ I'll rewrite the theorem 2.5 here for convenience: Let $\kappa$ be a singular cardinal of cofinality $\lambda$ which is regular in $K$. Then $\kappa$ is measurable in $K$ and if $\lambda>\omega$ then $o(\kappa)\geq\lambda$ in $K$. I'm not sure even what is $K$ here. $\endgroup$ Feb 20 at 12:40

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