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ZF is defined using axiom schemata, rather than a finite set of axioms. So ZF has an infinite (countable) set of axioms.

I realized that in my study of math I probably missed how matching a statement to be an instance of an axiom schema is defined.

We can't refer to the fact that schemata provides a set of axioms, because ZF and sets are not yet defined.

Please give a precise definition (e.g. a matching algorithm) for axiom schemata of ZF.

porton
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    I only see the most superifical resemblance between this qurstion and its supposed duplicate, and hence voted to reopen. – Z. A. K. Oct 05 '22 at 06:57
  • In particulat, @asaf-karagila's answer there does not explain the (admittedly fairly obvious) algorithm for recognizing instances of ZF schemata. – Z. A. K. Oct 05 '22 at 07:01
  • @Z.A.K. I don't see how a "precise definition" is the same as "computer algorithm". In fact, "e.g." means "for example", i.e., a computer algorithm is an example for a precise definition. – Asaf Karagila Oct 05 '22 at 07:10
  • See Schema: "For example, in first-order number theory the induction principle is specified using the schema $[F(0) \land ∀x((Num(x) \land F(x))→F(sx)]→∀x(Num(x)→F(x))$ where the two blanks marked ‘$F(x)$’ are to be filled with a first-order formula having one or more free occurrences of the variable ‘$x$’, the blank marked ‘$F(0)$’ is to be filled with..." – Mauro ALLEGRANZA Oct 05 '22 at 08:01
  • In a nutshell a "schematic variable" for a formula $\Phi$ or $\varphi$ must be instantiated with a syntactically correct formula of the language. – Mauro ALLEGRANZA Oct 05 '22 at 08:05

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