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Who first used the corner quotes, ⌜ and ⌝, or $\texttt{\Godelnum}$ with Sam Buss's macro, for the notion of Gödel number?

Quine introduced corner quotes, but did not use them for the notion of Gödel numbers.

The corner quotes were used for Gödel numbers by Joseph R. Shoenfield, Mathematical Logic, 1967, as from page 122.

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The Corner quotes are used prevalently in provability logic, and in other areas of logic. Two important citations are Craig Smorynski, The Incompleteness Theorems, in Handbook of Mathematical Logic, as from 1st edition 1979; and Paanu Raatikainen, Gödel's incompleteness theorems, in Stanford Encyclopedia of Philosophy.

Edit: I do not find the corner quotes in Martin Davies, The Undecidable, or in the books by Moztowski or Kleene.

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    $\begingroup$ I agree with your last edit: I've tried a check on a quite huge number of textbooks from '60s and I've found no similar use. So, I've no evidence when the original Quine's quotes, used in ML (1951, revised edition) in the context of Incompleteness but with a different meaning (see page 314), has been used for the "godel numbering function" before Shoenfield (if any). For example, Raymond Smullyan, Formal Systems (1961) uses $g(x)$. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Sep 27, 2022 at 9:56
  • $\begingroup$ @MauroALLEGRANZA Thank you for the research! You may offer an answer. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Sep 27, 2022 at 18:42
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    $\begingroup$ @MauroALLEGRANZA May you compare with the answer offered to the same question here? mathoverflow.net/questions/430869/… I will check the references provided there, tomorrow Brazilian time. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Sep 27, 2022 at 22:05

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