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It seems like there’s potentially a kind of subtle circularity in how the concept of countable (i.e. countably infinite) first-order languages is defined. On the one hand, one putative goal of first-order logic is to rigorously axiomatize the natural numbers. But on the other hand it seems like you kind of need the concept of natural number to already exist in order to define what a countable language is: arguably, recursive grammar definitions can mostly avoid direct mention of “number concepts”; but once you need to talk about arbitrary “arities” of relations and functions (like $n$-ary relation for “any $n$”) then “number concept” seems unavoidable.

Do logicians have an accepted way of resolving this?

  • I could hypothesize this: Imagine you’ve started with a finite “mini-language” (finitely many symbols and sentences), used the “mini-language” to define natural numbers, and then leveraged the now-defined natural number concept for the full-blown definition of the countable first-order language concept.
NikS
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  • While the duplicates are not about the concept of a natural number per se, the same principles apply here. – Asaf Karagila Apr 07 '23 at 06:42
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    The goal of most logicians / logic textbooks in defining first-order arithmetic is not to define numbers ex nihilo. It's to formalize reasoning, so it can be made an object of study, and so e.g. we can have a reasonably clear standard on which proofs are valid proofs (ones that coupd in principle be formalized). As such, the books explain it in terms of concepts they assume you already know, such as numbers. NB Some folks will confidently assert that eliminating this "circularity" is impossible anyway, but even those who disagree with that are rarely interested in attempting such a project. – Z. A. K. Apr 07 '23 at 06:51
  • You might want to read about Logicism, a philosophical effort which does aim to ground some or all of mathematics in logical concepts and intuitions -- nb a more modest aim than defining specifically FOL without any reference to numbers, yet one not regarded by most philosophers as entirely successful at this stage. But keep in mind that the vast majority of logicians and logic textbooks do not have logicist goals or purposes, and will not aim at all to eliminate dependence on arithmetical concepts, not even at the level of "lip service". – Z. A. K. Apr 07 '23 at 07:03
  • So @Z.A.K. , if formal logic doesn’t aim to define natural numbers ex nihilo (and in fact assumes they have already been defined in some sense) what is the purpose/utility of the Peano axiomatic definition in the language of 1st or 2nd order logic? This definition seems redundant if the language assumes natural numbers have already been defined. Maybe the purpose (even if the axioms are “redundant”) is to lay the foundation for using formal logic to reason *about* natural numbers? – NikS Apr 07 '23 at 08:59
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    @NikS: There are countless uses of FO arithmetic. Common goals are to study either proofs about arithmetic (e.g. can we find simple reasoning principles that allow us to derive most other principles we use?); get computers to check our proofs; study un/decidability phenomena (there is no algorithm deciding arithmetic truth, quantifiers cannot be eliminated); investigate models of various fragments of arithmetic; prove results about whether set-theoretic assumptions (e.g. Continuum Hypothesis) affect arithmetic at all; as a mere technical tool; etc. And this is just the tip of the iceberg. – Z. A. K. Apr 07 '23 at 10:40
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    @NikS: All of these investigations are made easier by having a first-order theory of arithmetic - and a good logic textbook will introduce at least some of these topics. Notice that these questions are of independent mathematical (and often pragmatic) interest even if (in fact, especially if) you already have a good rigorous understanding of numbers, and you have no concerns about their rigorous justification at all. – Z. A. K. Apr 07 '23 at 10:43

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