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These two theories are known to be first-order theories. And the definition of the first-order logic typically involves something like "set of symbols $a_1, a_2, a_3...$", which already includes set and numbers. And also the proofs by induction are also always accepted without any underlying rigorous reasoning.

So, ZFC and Formal arithmetic and their corresponding intuitive analogues are basically two completely different things, one built using the other, not the different ways to represent the same object.

Since the answer to my question is most likely just "yes", I'll expand it a little bit:

Is there any standard conventional "list of primitive notions", such as collection(as informal set), number etc. that all the logic is built upon?

Sgg8
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  • Yes, they are built using natural language that already uses the intuitive processes of counting and "collecting". – Mauro ALLEGRANZA Aug 18 '22 at 09:55
  • Regarding induction, it's not"without any underlying rigorous reasoning". It can be justified via (rigorous) meta-reasoning, even though it still relies on some prior acceptance of properties of ℕ or finite symbol strings. – user21820 Jul 04 '23 at 08:40
  • @user21820 by "rigorous", I actually meant "not accepted as a meta-fact" – Sgg8 Jul 04 '23 at 14:55
  • Then yes, but it shouldn't be surprising. The only reason we came up with PA in the first place is because there appears to be some structure in the real world that looks like it satisfies PA. So the only justification of PA is by its connection to reality. The induction schema of PA is justified precisely by that connection as well, since every proof is a finite sequence of symbols in.. reality! – user21820 Jul 06 '23 at 06:24

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Is there any standard conventional "list of primitive notions"

If you choose ZFC as your foundation, the only primitive notion is set. And than everything else is to be expressed with sets accordingly to the axioms. Say, natural numbers {}; {{}}; {{}, {{}}} for 0, 1, 2 and so on.

And also the proofs by induction are also always accepted without any underlying rigorous reasoning.

Proofs by inductions don't have to be intuitive, they also can be rigorously reasoned, but in the end will necessarily refer to certain axioms (e.g. the axiom of infinity in ZFC).

Hopefully it will help, but I might be wrong at the matter, sorry, if so.

swcoll
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    Using ZFC as a foundation doesn't seem to be a good idea, since ZFC is a first order theory and in order to explain what it is, you'll have to disregard formal theories and confine yourself to a way smaller world. And you're still going to need to use something like "set of axioms", "set of free variables" for ZFC – Sgg8 Aug 20 '22 at 04:05