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I know the first-order version of ZFC, but not second-order ZFC. Can anyone explain how axioms (and other things) of second-order ZFC differ from the first-order version?

Trevor Wilson
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hwe
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Second-order logic allows us to quantify over subcollections of the universe. This allows us to write things like "For every class which contains all the ordinals ..." within the theory, whereas in first-order logic these sort of sentences are actually schema of sentences (where class becomes a definable class).

It is important to point out that there are two [common] ways to interpret second-order logic, the first is with Henkin semantics where the classes are allowed to be definable classes only; this results in something resembling first-order logic. The second way is to full semantics which allows the classes to be any collection.

In second-order ZFC instead of the replacement schema we have one axiom which says that every function whose domain is a set then its range is a set.

For example we don't have countable models of second-order ZFC when interpreted in full-semantics, because those "hide" a very [first-order] undefinable bijection between $\omega$ and their universe. Second-order replacement can see this bijection, so the model cannot satisfy replacement (The range of a function whose domain is a set is everything).

Asaf Karagila
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    This is not a very thorough explanation... There is no mention of the difference in the language, and in the second paragraph full second-order semantics are being assumed, while second-order ZFC could also be studied in Henkin semantics. – Carl Mummert Feb 22 '13 at 11:42
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    Isn't second-order ZFC in Henkin semantics logically equivalent to first-order ZFC? – Zhen Lin Feb 22 '13 at 11:44
  • @Carl, yes you are correct. I didn't write a long answer for two reasons: it seems the OP didn't try to search on their own first, and more importantly I am writing from my iPhone. :-) – Asaf Karagila Feb 22 '13 at 11:44
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    @Zhen Lin: depending on which comprehension and choice axioms are added for the classes, it could end up being equivalent to NBG, or to MK, or to one of these with additional choice principles. The need to add additional comprehension in order to obtain a theory that interprets ZFC is another key detail to mention. – Carl Mummert Feb 22 '13 at 11:49
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    @Carl: I should also add that this is not the first time you are trying to prod me into improving my answer instead of posting your own, and I am certain that many of these times you could have given a much better answer. – Asaf Karagila Feb 22 '13 at 11:51
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    @Carl: Now by a computer I added a bit (and correct an iPhone generated typo, and made room for man-made typos instead). I still think that you should write something yourself as well. – Asaf Karagila Feb 22 '13 at 12:37