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Take the axiom of Comprehension that was introduced by Zermelo. This, as usually stated has one instance for every single formula.

Presumably this is to contain the axiom within first-order logic.

Still, as a technical manuevre it seems something of a hack. Unless of course there is a deeper motivation behind it, or that it signals something deep in the structure of first-order ZFC. Does it?

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You are right; the limited expressive capabilities of first-order alnguage impose the use of axiom schema, like Separation and Replacement in ZF (Zermelo-Fraenkel system) and Induction in (fist-order) Peano arithmetic.

Wherever you need to quantify a predicate (an "attribute"), in first-order logic you must use a schema, with the limitation that while (with standard semantics) attributes in a countable domain are uncountable (i.e. if the domain of individuals has "aleph_0" elements, the possible attributes are as many as all the possible subsets, i.e. have the cardinality of the power-set of the domain), the possible "instantiations" of a schema are countable (we have only countable many formulas in the language).

But, regarding set theory there are more than one axiom system : see in SEP Alternative Axiomatic Set Theories; specifically the NGB system (von Neumann-Godel-Bernays) is finitely axiomatizable:

The resulting theory is a conservative extension of ZFC: it proves all the theorems of ZFC about sets, and it does not prove any theorem about sets which is not provable in ZFC. [...]. two comments about this. First, the mental furniture of set theorists does seem to include proper classes, though usually it is important to them that all talk of proper classes can be explained away (the proper classes are in some sense “virtual”). Second, this theory (especially the version with the strong axiom of Limitation of Size) seems to capture the intuition of Cantor about the Absolute Infinite.

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Its interesting that NGB is finitely axiomatisable and includes proper classes. I'm not sure about the SEPs comment about what Cantor meant by the absolute infinite - I think he might have been more subtle about it. Why isn't then NBG canonical rather than ZFC? – Mozibur Ullah Feb 19 '14 at 10:17
Possibly ZFC being first past the post with a workable formalised set theory was enough to trump NBG. – Mozibur Ullah Feb 19 '14 at 10:34
@MoziburUllah - Elliott Mendelson's textbook (Introduction to Mathematical Logic, 4th ed - 1997) follows NBG. The discussion about Cantor's Absolute Infinite point at the fundamental book of Michael Hallett, Cantorian set theory and limitation of size (1986). See also in SEP the entries on The Early Development of Set Theory by José Ferreirós, and Zermelo's Axiomatization of Set Theory, by Michael Hallett itself. – Mauro ALLEGRANZA Feb 19 '14 at 10:40

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