# Changing the cofinality of a regular cardinal without collapsing any cardinals?

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?

• I believe this question is coming from a non-expert in set theory looking for advice from people who are experts. Feb 18 at 10:56

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.
• 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 ? Feb 20 at 12:25
• 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. Feb 20 at 12:40