Some time ago, Asaf Karagila wrote wonderful post wherein he shows that, even without assuming the axiom of choice one can always find four cardinals \(\mathfrak{p} \lt \mathfrak{q}\) and \(\mathfrak{r} \lt \mathfrak{s}\) such that \(\mathfrak{p}^{\mathfrak{r}} = \mathfrak{q}^{\mathfrak{s}}.\) In the comments, Harvey Friedman asks:

Your theorem is an example of an existential sentence about cardinals in the language with only \(\lt \) and exponentiation. Can you determine which sentences in that language are provable in ZF? More generally, expand the set of sentences about cardinals considered, obviously to include addition and multiplication, and perhaps alternating quantifiers.

After Peter Krautzberger’s tiny blogging challenge, I decided to spend 30 minutes thinking and writing about this…