I also vote for ZF set theory. That screenshot is not really
representative -- it was just a quick example that I thought illustrated
a simple argument that was made even more transparent in the formalization.
I've talked a bit with some of the Isabelle developers about different
ways of importing Isabelle/HOL (and even HOL Light or HOL4) results
into Isabelle/ZF, but I entirely agree with you that Isabelle/ZF (in a
human-readable Isar style) is the correct basic choice.
It would be nice to eventually figure out how to make those aspects
of mathematics which are more easily done in HOL
(as Paulson has recently mentioned:
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2008-September/msg00039.html )
equally easy in ZF. Perhaps locales or other methods of abstracting away
the low-level defintions may be useful here.
It's also unclear to me to what extent Isabelle/HOL is an easier
setting in which to do certain sorts of math due to the type theory
vs. just a larger existing body of results. The latter will naturally
improve over time, but it's worth seeing how we might mitigate the former.
Best,
- Cameron