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I really should know this, but ...

When/where/by whom was first-order Peano Arithmetic first clearly and explicitly formulated in a recognizably modern form (perhaps exact notation apart) -- with the usual recursion axioms for successor, addition, multiplication and the schema for induction axioms?

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    $\begingroup$ But the question is whether they are "first order". As I understand it, Peano's formulation was "second order", since the induction principle was formulated for (in modern language) "all subsets", not merely for subsets expressible as a formula. $\endgroup$ – Gerald Edgar Dec 8 '20 at 12:09
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    $\begingroup$ It seems that you can find Hilbert's notes into David Hilbert, Lectures on the Foundations of Arithmetic and Logic (William Ewald & Wilfried Sieg editors, Springer (2013)), page 925 (Hamburg 1927 lecture) and page 979 (Hamburg 1930 lecture). $\endgroup$ – Mauro ALLEGRANZA Dec 9 '20 at 15:57
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    $\begingroup$ @MauroALLEGRANZA It looks like Hilbert/Bernays (so probably Bernays!) is the earliest. I guess it has to be post-Gödel 1931 that people settled on a first-order formulation with just successor, addition, multiplication built in, because it was only after we know the beta-function trick that we know we don't need separate explicit definitions for the other primitive recursive functions (which come for free in the second-order setting). $\endgroup$ – Peter Smith Dec 11 '20 at 12:00
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    $\begingroup$ @PeterSmith - I agree. Gödel (1931) is till Principia-like: high-order logic as underlying logic but instead of W&R "mathematical" axioms (Reducibility, Infinity) the SOL version of Peano's axioms. As you said, the "prominence" of FOL emerged slowly. $\endgroup$ – Mauro ALLEGRANZA Dec 11 '20 at 12:06
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    $\begingroup$ Two books I have (but don't know enough nor have time now to dive into) that might be worth looking through are Perspectives on the History of Mathematical Logic edited by Thomas Lyndon Drucker (1991) and The Search for Mathematical Roots. 1870−1940 by Ivor Grattan-Guinness (2000). The second almost certainly has everything and more that you want, but trying to distill exactly what you want, and no more, is probably going to be very difficult (which I'm sure you know if you're familiar with G-G's book). $\endgroup$ – Dave L Renfro Dec 11 '20 at 19:53

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