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.
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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.
Noah Schweber
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