Adding new bijections without adding new subsets by forcing

Question

I found this post very interesting. It shows that, by forcing, any given cardinal $\kappa$ of the ground model can be made countable in a forcing extension by adding a bijection between $\omega$ and $\kappa$.

Now, I am curious, can this be done without adding new subsets to $\kappa$?

Thank you in advance!

Answer 1

It can't even be done without adding new subsets to $\omega$. If $f:\omega\rightarrow\kappa$ is (in the generic extension $V[G]$ in which $\kappa$ is made countable) a bijection, let $R\subseteq\omega^2$ be the pullback along $f$ of the usual ordering on $\kappa$; that is, $R=\{(a,b): f(a)<f(b)\}$. Via your favorite pairing function on $\omega$, we can then code $R$ as a set $r\subseteq\omega$. But from this $r$ we can recover $f$ itself, so $r\in V[G]\setminus V$.

Answer 2

No, of course not.

Any function from $\kappa$ to itself, let alone a function from a smaller ordinal, is a subset of $\kappa\times\kappa$. Since we have a canonical bijection $\kappa\times\kappa\to\kappa$, any new function must be a new subset.