The minimal transitive model of $\sf ZF$ as an arbiter on Platonic truths of independent statements of $\sf ZF$?

Question

Statements in the language of set theory that are independent of $\sf ZF$, and that are within the consistency level of $\sf ZF$, like $\sf AC$, $\sf CH$ and $\sf GCH$, etc... These need a certain truth criterion to decide on whether they are true or not in let's say the Platonic world of sets, which we would take it to be a single world (not a multiverse).

Why we don't take the decision made by the minimal transitive model of $\sf ZF$ on those statements to be that criterion?

I'm asking about the mathematical side of this question, I mean in what sense this arbiter is poor or otherwise strong? Are there known interesting independent statements that it cannot decide on?

Answer 1

The problem with the minimal transitive model is that it's $L_\alpha$ for some countable ordinal $\alpha$.

This model is very poor in large cardinals, indeed it has none, and if we want to think about set theoretic background for forcing, and we want to use countable transitive models for that, then this is again not a suitable choice, since it will not have any transitive models of $\sf ZFC$ inside of it.

One can argue that adopting $V=L$ + "No transitive models of $\sf ZFC$" will answer the vast majority of our reasonably independent questions, like $\sf GCH$, like the existence (or rather lack thereof) of large cardinals, etc. But it does so in an incredibly poor and boring way.

Besides, unlike with the case of $\sf PA$, where "the standard model" is somehow tangible and understood, this $L_\alpha$ is still fairly abstract and $\alpha$ is incomprehensibly large as far as countable ordinals go. After all, it will know all the reasonable things you can do with Turing machines, their oracles, and so on. So it is necessarily closed under all the standard recursive hierarchies and all that.

Answer 2

Based on your comments, it sounds like you want to focus on questions either answer to which adds no consistency strength to $\mathsf{ZF}$, and for such questions set a "default" truth value based on $L_\alpha$. There are various possible responses, but one I think is important to give is that the interaction between such "simple" questions and high-consistency-strength principles is very complicated in general.

For example, we're used to the idea that large cardinals - even very large cardinals - don't impact the continuum function. But that takes serious work; I don't believe that there is a trivial argument that (for example) if the existence of a supercompact cardinal is consistent with $\mathsf{ZFC}$ then it is consistent with $\mathsf{ZFC+GCH}$. Certainly I'm not comfortable turning to $L_\alpha$ for answers to "natural combinatorial questions" (even assuming we can somehow dodge those secretly referring to $L$ itself) and blithely assuming that the answers so provided do not impinge on other aspects of the set-theoretic universe that are important to me.