Before asking this question, I want to know if anyone has formalized what an axiom schema is. Assuming that there is a formalization, many theories we normally encounter have a finite number of axiom schemas, even if they are not finitely axiomatizable. Examples would be the propositional logic and ZFC set theory and Peano arithmetic. But is there a recursively axiomatizable theory that can't be axiomatized with a finite number of schemata? I would at the very least like to know if anyone else has thought about these kind of things, and perhaps formalized the notion of a set of axioms that is a schema.
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