I sometimes see books just having "axiom of specification" rather than lengthy "axiom schema of specification". Are these two the same thing?
2026-03-30 16:54:30.1774889670
Are axiom schema of specification and axiom of specification the same terminology?
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Yes; "axiom of specification" (or separation) is shorthand. It's technically incorrect, but clear.
. . . Unless you're working with second-order set theory! Then you can collect the whole scheme into one axiom. This is extremely rare, though, and almost without exception this isn't what people mean. But - especially if you're reading about old set theories - you may run into this.
Note that historically, the original presentation of Zermelo set theory was second order (well, actually, if I understand correctly it was informal, not even second-order; but I believe our historical understanding is that Zermelo meant the second-order version).