By Peano Arithmetic I mean first order Peano Arithmetic. The earliest proof that it is not finitely axiomatizable that I know of is R. Montague, Semantical Closure and Non-Finite Axiomatizability I. J. Symbolic Logic 29 (1964), no. 1, 59--60. But was the result known by other means before that?
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In 1952 Czesław Ryll-Nardzewski proved that first order PA is not finitely axiomatizable. The proof uses nonstandard models. Andrzej Mostowski proved the same result (also in 1952) but without using nonstandard models.
Thorsten Elof
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7The Mostowski reference is: Andrzej Mostowski, 1952, "Models of axiomatic systems", Fundamenta Mathematicae 39:1, 133-158. See sec 7 of http://matwbn.icm.edu.pl/ksiazki/fm/fm39/fm39112.pdf . Mostowski simply demonstrated the now-standard fact that PA proves the consistency of each of its finite subtheories. – Carl Mummert Nov 11 '17 at 21:02
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Czesław Ryll-Nardzewski, The Role of the Axiom of Induction in the Elementary Arithmetic, Fundamenta Mathematicae 39 (1952).
Mauro ALLEGRANZA
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1Thanks. This paper manipulates non-standard models to show every sentence of PA (and thus every finite conjunction of axioms) has a model where some instance of the induction axiom scheme fails. – Colin McLarty Nov 11 '17 at 21:04