Preservation of cardinals implies preservation of cofinalities when $V=L$?

Question

Kunen mentions the result stated in the title. It would be much appreciated if someone can give a reference for this. Thank you!

Answer 1

I presume you mean that if $L[G]$ is a set forcing extension of $L$ with the same cardinals, then they have the same cofinalities. This follows easily from Jensen's Covering Lemma. That can be seen in Jech (3rd millenium ed). (Suppose $L[G]$ has some cardinal $\kappa$ which is regular in $L$ but singular in $L[G]$; let $\mu=\mathrm{cof}^{L[G]}(\kappa)<\kappa$. So $\kappa$ is a limit cardinal in both $L$ and $L[G]$. Let $X\in L[G]$ be such that $X\subseteq\kappa$ and $X$ has ordertype $\mu$. Then by the Covering Lemma, there is $Y\in L$ such that $X\subseteq Y$ and $\mathrm{card}^{L[G]}(Y)\leq\max(\aleph_1^{L[G]},\mu)$. But we may assume $Y\subseteq\kappa$, and since $Y\in L$, where $\kappa$ is regular, $Y$ must have ordertype $\kappa$, so $Y$ has cardinality $\kappa$ in $L[G]$, contradiction.)