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My understanding is that the ZFC axioms are written in the language of first-order logic. However, it seems like one needs the ZFC axioms (or some other set theory axioms) to define first-order logic.

For some examples:

  • The axiom of infinity is used for the existence of an infinite set of symbols which we call variables.
  • The axioms schema and pairing are required to show the existence of Cartesian products. And relations, proofs, well-formed formulae etc. are all elements of certain Cartesian products.
  • The axiom schema is seemingly used to define what a well-formed formula is, as well as what a proof is.

And the list goes on. Do the sets which are used to define first order logic actually come from the ZFC axioms? If so, this seems like a circular system; one needs first order logic to define the ZFC axioms, but to define first order logic one needs the ZFC axioms.

What is the solution to this conundrum? Any help is appreciated.

Austin Shiner
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  • Don't take my word for this, I am not an expert. Taking from the words of my instructor in grad set theory/logic, this is a problem. Set theory is defined using first-order logic, and first-order logic is defined via set theory. This is one reason why the "foundational crisis" as it's so-called is still going on. – Moni145 Jul 24 '21 at 01:49
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    Only a very weak fragment of set theory is needed to justify the standard definitions of first order logic. Most introductory texts on logic explain this in terms of languages comprising finite sequences of symbols. I think you should cite the books that are giving you difficulties. – Rob Arthan Jul 24 '21 at 02:16
  • @RobArthan A finite sequence of symbols is an element of a Cartesian product, which as far as I know uses both the axiom pairing and axiom schema to construct no? Those seem like pretty non-trivial axioms. Unfortunately there is no particular book I am going off of; I learned set theory and first-order logic in undergrad lectures at my university and this is just something I've been thinking about. – Austin Shiner Jul 24 '21 at 04:17
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    You can't get started in logic if you don't accept some principles for dealing with syntax.Those principles are necessarily prior to abstract concepts like pairs in set theory. – Rob Arthan Jul 24 '21 at 04:32
  • @RobArthan Yes, I guess a reformulation of my question is what exactly do we need to assume in order to define first-order logic? Or in other words, what symbols/concepts do we just assume to have mathematical meaning? For instance do we assume that certain finite strings of symbols have meaning, and if so how do we differentiate it from strings in the set theory sense? – Austin Shiner Jul 24 '21 at 05:21

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