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Reading Skolem's 1923 Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlicher mit unendlichem Ausdehnungsbereich (Foundation of elementary arithmetic through the recurrent method of thought without use of apparent variables ranging over an infinite domain), I was wondering just how much of arithmetic Skolem's method can actually represent?

His stated goal is to show that: "if one regards the general propositions of arithmetic as functional assertions, and if one bases oneself on the recurrent method of thought, then this science can coherently be given a foundation without application of the Russell-Whiteheadian concepts 'always' and 'sometimes'" (my sketchy translation). In other words, Skolem wants to represent arithmetic in a logical system that makes no use of unbounded quantification, but which allows what he calls "die rekurrierende Denkweise", i.e. recursive definition.

But in what way can Skolem claim that his is a foundation of arithmetic that dispenses with unbounded quantification? For he uses the principle of complete induction in most of his proofs, and thereby (in my view) smuggles quantification over all natural numbers back in. If Skolem were true to his project to exclude unbounded quantification, how much of arithmetic could he actually represent?

I have a more general question, but I'm not sure that I am phrasing it correctly: is it possible to define the logical quantifiers in terms of more primitive notions? For example, I know that in constructivist approaches to the semantics of logic, the meaning of universal quantification and implication are analyzed in terms of the notion of a constructive function. Think of the universal quantifier introduction rule in a Gentzen system,

$$ \frac{A(a)}{\forall x A(x)}, $$ where the proof of $A(a)$ does not depend on the choice of $a$ and therefore provides us with a general method to show that $A$ holds universally. Universal quantification is here analyzed in terms of the concept of a (constructive) function, and it seems that this should be sufficient to account for all cases of quantification we might come accross say in Heyting arithemtic. Would it not be possible in this way to give the kind of account that Skolem was after, a foundation of arithmetic in terms of functions (defined in some way according to recursion theory).

This opens the wider question: in what sense can we count recursion theory to logic? Skolem evidently thought that the "rekkurierende Denkweise" is a purely formal feature of logical thought, that in some sense, other logical notions such as quantification can be replaced by it, and that this would be a less problematic way of founding mathematics than by appeal to quantifiers ranging over infinite domains.

Rodrigo
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  • I don't think this question is clear enough to be answered here, but you're welcome to [the logic chat-room](https://chat.stackexchange.com/rooms/44058/logic) for informal discussion! =) – user21820 Nov 19 '18 at 17:32
  • For translations, you may see "FROM FREGE TO GODEL A Source Book in Mathematical Logic, 1879-1931". For the power of complete induction, take this with sketicism but I believe Gentzen's proof of the consistency of arithmetic already discusses the use of complete induction. I think he argues why this is fine from a constructivist point of view, but don't check me on that. – Rodrigo Dec 30 '21 at 14:57

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