Permutation model in which infinite sets are Dedekind-infinite

Question

I've been looking for a permutation model in which $IDI$ (every infinite set is Dedekind-infinite) and $\neg AC$ holds. I have found several (for example in Countable Sums and Products of Metrizable Spaces in $ZF$ by Keremedis and Tachtsis, Theorem 14), but they all require a lot of group theory and very long proofs to show that $IDI$ holds. I was wondering if anyone is aware of an example of such a permutation model with a much simpler proof that $IDI$ holds?

Answer 1

The simplest examples are in Chapter 8 of Jech's "Axiom of Choice" book.

Simply put, it is not hard to verify that if the filter of groups is closed under countable intersections (or if the filter is generated by fixing an ideal of sets, then the ideal is closed under countable unions), then $\sf DC$ holds in the model.

So, for example, when the atoms have size $\aleph_1$, taking all the permutations of the atoms and fixing pointwise any countable set of atoms will do the trick.