The axiom schema of induction in Peano Arithmetic that I read about in Wikipedia concerns a tuple of free variables $(x, y_1,…, y_k)$. My question is whether more than $x$ is necessary. In other words, would the theory be strictly weaker if we only allowed one free variable in the axiom schema of induction. A similar question can be asked in the axiom schemas of comprehension and replacement in ZFC.
Is more than one free variable necessary for the axiom schemas in Peano arithmetic and ZFC set theory
logic
Best Answer
We can indeed drop parameters from the induction scheme in PA; see e.g. here.
The situation is the same for ZFC; see this paper of Schindler and Schlicht.
However, both of these facts are nontrivial, and in general parameter-free systems tend to be much weaker than their fully-parameterized counterparts. So I'd personally argue that even though we can omit parameters in the presentations, we shouldn't.