I encounter this issue when going through equivalent characterizations of measurable cardinals. For completeness, let me reproduce the statement:
For ordinal $\kappa$, the following are equivalent.
- There is a transitive class $M$ and an elementary embedding $j:V\to M$ with $crit(j)=\kappa$.
- $\kappa>\omega$ and there is a non-principal $\kappa$-complete ultrafilter on $\kappa$ (i.e. a measure).
- $\kappa>\omega$ and there is a normal measure on $\kappa$.
The way I interpret it is that we are trying to find a first-order set-theoretic characterization of (1) so that we don't have to ever mention elementary embeddings and can safely work in our usual universe.
The proof of $2\implies 1$ uses ultrapower construction, and along the way we are trying to show its well-foundedness by assuming a decreasing chain $\{[f_{i+1}]\dot{\in}[f_{i}]:i<\omega\}$, picking an $f_i$ to take their intersection $\bigcap_{i<\omega}\{\alpha<\kappa:f_{i+1}(\alpha)\in f_{i}(\alpha)\}\in U$ to reproduce a decreasing chain $f_{i+1}(\alpha)\in f_{i+1}(\alpha)$ and contradict foundation in $V$.
Now, here is my question: visibly, this proof chooses a representative out of each equivalence class. But what if we don't even have weak forms of choice in our universe (not even, say, countable choice)? Is there still some way to get around it, or do we just give up? Or am I missing something? (I am new to large cardinals and so may be careless when going in between models and universes; please be patient :D)
Best Answer
In $\sf ZF$, (1) implies (3) implies (2) and none of those are reversible. The forward implications are easy and the usual proofs are choice free.
The failure of (2) implies (3) is a fairly involved proof due to Bilinsky and Gitik. The failure of (3) implies (1) is easy from models where $\omega_1$ is measurable, as it is never the critical point of a definable embedding.
One of the works of Mitchell Spector involved studying the choice principles needed for the equivalence between the three statement, and this turns out to be "every family of non empty sets indexed by $\kappa$ has a subfamily indexed by a measure-1 set".