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.
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7$\begingroup$ The Mostowski reference is: Andrzej Mostowski, 1952, "Models of axiomatic systems", Fundamenta Mathematicae 39:1, 133-158. See sec 7 of 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. $\endgroup$ Nov 11, 2017 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).
answered Nov 11, 2017 at 20:51
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1$\begingroup$ Thanks. 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. $\endgroup$ Nov 11, 2017 at 21:04