What is the definition of the Feferman-Levy model?

Question

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.

Answer 1

Yes, the Feferman-Levy model is described in Jech The Axiom of Choice in Chapter 10 (Theorem 10.6), as well you can find nice detailed accounts in both Arnie Miller's papers about Dedekind-finite Borel sets and Long Borel Hierarchies (both of which you can find here), as well Ioanna Dimitriou M.Sc. and Ph.D. thesis (both of which appear here).

The idea is that we begin with $L$ (at least in the classical settings, we don't have to in general), then we use the forcing $\Bbb P=\prod_{n\in\omega}\operatorname{Col}(\omega,\omega_n)$, and take all the names which are fully decided in a finite stage of the product.

This can be given a nice symmetric extension presentation as well. Note that conditions in $\Bbb P$ are finite functions accepting pairs of integers, and $p(m,n)<\omega_m$. Now take permutations which move for each $m$ separately, then $n$ coordinate. Then take the filter of subgroups of those fixing finitely many $m$'s.