[FOM] The origin of second-order arithmetic

[Ali Enayat]

This is a reply to a query of Sam Sanders (03 Dec 2017) who expressed
his reasons for doubting that SOA was originally formulated by Hilbert
& Bernays in their Grundlagen der Mathematik.

One of the earliest key researchers on second order arithmetic (SOA)
was Andrzej Mostowski, who I have found to be a reliable source of
references, and judicious with giving credit to his predecessors. So I
consulted the following paper to see what says:

On models of axiomatic systems. Fund. Math. 39 (1952), 133–158 (1953).

The above paper is the earliest paper on SOA by Mostowski that is
referenced in MathSciNet .

In the above paper Mostowski proves many results, including the
non-finite axiomatizability of  including Zermelo set theory, SOA (in
section 5, where he refers to SOA as "Axiomatic Theory of Real
Numbers"), and first order arithmetic.

Mostowski's paper makes several references to earlier papers and books
(including those by Church and Hilbert-Bernays) in connection with set
theory and first order arithmetic, but curiously in the section
dealing with SOA in section 5 of the paper, he does not provide any
references whatsoever.

This suggests that perhaps this paper of Mostowski is the first time
that SOA saw the light of day, even though it may have been in the air
for quite some time.

Best regards,

Ali Enayat