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I was reading an introduction of ZFC set theory I found: https://ia803008.us.archive.org/31/items/A_C_WalczakTypke___Axiomatic_Set_Theory/Lecturenotes2006-2007eng.pdf

Chaper 1 covers the idea of "formula" and "sentence" then stated the idea of formal deduction:

Formally, we define S ⊢ φ iff (= ”if and only if” =⇔) there is a formal deduction of φ from S. That is, iff there is a finite sequence φ_1, . . . , φ_n of formulas such that φ_n is φ, and for each i, either φ_i is in S, or φ_i is a logical axiom, or φ_i follows from φ_1, . . . , φ_{i−1} by certain rules of inference.

Now I'm curious about what the "logical axiom" and "certain rules of inference" are or should/could be (in set theory). But some quick search I only found those that are either vague, far from ZFC or too complicated for me to grasp.

I did find this post, the question being

What are the "standard" inference rules for set theory?

which is close to my question, though I found the comments a bit confusing(and the post has no "answer").

Q: Originally, I want to ask for a list of logical axioms and rule of inference for set theory, and maybe an example of formal deduction using the rules of inference. But now also I want to ask about the first comment in that post and why is it related.

See First order logic : usual Hilbert-style axioms with Modus Ponens.

Any books or other sources are appreciated. Thanks in advance.

J.-E. Pin
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Lab
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  • ZFC is a first order theory. The logical axioms and rules of inference for ZFC are exactly the same thing as the logical axioms and rules of inference for first order logic. So yes, a general reference to deductive systems for first order logic is appropriate. – spaceisdarkgreen Sep 10 '23 at 14:42
  • @spaceisdarkgreen I see what they meant, thanks. May you give an example of any (simple) deductive system for first order logic to actually prove some theorem (sentence) in set theory based on the ZFC axioms? – Lab Sep 10 '23 at 15:20

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In the comments, you clarified your question:

May you give an example of any (simple) deductive system for first order logic...

Well, you can open up literally any introductory logic textbook. An example of such a system (which is called a natural deduction system) is described on wikipedia here.

...to actually prove some theorem (sentence) in set theory based on the ZFC axioms?

In my answer here, I wrote out all the details of a natural deduction proof that the empty set is unique.

Alex Kruckman
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