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In the Wikipedia article on General Set Theory (GST), there is a claim in the section on metamathematics that seems to be missing a disjunct (or shouldn't be a disjunction at all):

GST is:

  • Not finitely axiomatizable. Montague (1961) showed that ZFC is not finitely axiomatizable, and his argument carries over to GST. Hence any axiomatization of GST must either include at least one axiom schema such as Separation; [emphasis added]

My question is what is the other disjunct such that GST must "...either include at least one axiom schema such as Separation..." OR ____? Is it something like "...or the second-order closure of such a schema"? Something else? Unclear and safe to ignore? (Or is the "either" just a mistake?)

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The "either" is likely a typo. An axiom schema is a rule for constructing an infinite number of axioms, all of which must have a given form. Not "finitely axiomatizable" means exactly what it sounds like. The set of consequences of the theory are not provable from a finite set of axioms. So here what it is saying is that it needs at least one axiom that is actually an infinite set of axioms. Sometimes the definition of "finitely axiomatizable" is actually taken to be a theory that can be axiomatized without the use of an axiom schema.

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  • $\begingroup$ Thanks! That’s what I suspected — essential use of schema — but lack of familiarity and curiosity had me second-guessing. $\endgroup$
    – Dennis
    Jan 19 '18 at 3:33

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