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I'm relatively new to formal logic. I have found a list of ZFC axioms on Wikipedia, but do not know what the rules of inference are. Is there a resource for what these inference rules are, or could someone list them for me? I have a copy of Mendelson's text on formal logic, so I can consult that for further reading once I know what ZFC's inference rules are.

  • Well, $\sf ZFC$ is extralogical. The inference rules are logical. These are different parts of the system. – Asaf Karagila Jan 19 '14 at 20:17
  • I do not know what extralogical means, unfortunately. Which inference rules should I use to prove theorems in ZFC? Or is there something I am missing? –  Jan 19 '14 at 20:18
  • The logic is the underlying "CPU" and $\sf ZFC$ is the assembly code in which we write. Extralogical means that it's not a part of the CPU itself, so to speak. The inference rules are part of the underlying CPU, and $\sf ZFC$ is part of the program. – Asaf Karagila Jan 19 '14 at 20:20
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    http://en.wikipedia.org/wiki/First-order_logic – Carsten S Jan 19 '14 at 20:23
  • http://en.wikipedia.org/wiki/List_of_rules_of_inference - Does that contain all of the ones I would need? –  Jan 19 '14 at 20:37

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ZFC is a first-order theory with a classical first-order logic with identity. If textbooks sometimes don't specify a particular system of logic for ZFC it is because it doesn't matter so long as it is a full classical system. You could use the ugly system in Mendelson's book, a natural deduction system, a sequent calculus, a tableaux system ... it won't matter because if there is a proof from the ZFC axioms to the conclusion $\phi$ in one system, there will be a proof in the others. So just assume your favourite presentation of first-order logic is in play, and off you go ...

Peter Smith
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