Does the Axiom schema of specification only ensure a countable number of subsets?

Question

The axiom schema of specification allows us to take a set $S$ and a predicate $\varphi$ written in the language of ZF(C) and create a subset of $S$. However there can only be at most a countable number of predicate sentences, if you didn't have the powerset axiom would you be restricted to just definable (perhaps compatible) subsets?

Answer 1

It's true that ZF without the power-set axiom doesn't prove the exist of uncountable sets (for a transitive model, look at $L_{\omega_1^{\,L}}$ or at the set of all hereditarily countable sets), but the situation is more complicated than you're suggesting, for two reasons:

1. It's not quite true that the axiom schema of specification (or separation) only covers countably many subsets, because it allows for the use of parameters. This isn't a critical point, though, because it still wouldn't let you break out of the countable sets into the uncountable (although once you have an uncountable set, the axiom lets you generate uncountably many subsets, via the use of uncountably many different parameters).

2. This objection is critical: You can't define truth in set theory, by Tarski's theorem, so you can't enumerate the sets produced by the axiom schema within ZF. There's no way within ZF to map definitions in general to the sets that they define, so even with only countably many definitions, you can't conclude that there are only countably many definable sets.
    
   In fact, in the minimal transitive set model of ZF, every set is definable without parameters, but the model clearly doesn't think that there are only countably many sets.