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Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro.

They were used by Joseph R. Shoenfield, in Mathematical Logic, 1967, as from page 122.

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: The corner quotes $\ulcorner\urcorner$ were indeed first used by Quine, and long before Shoenfield, but not for the notion of Gödel number.

Edit 2: I do not find the corner quotes in Martin Davis, The Undecidable, or in the books by Moztowski or Kleene.

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    $\begingroup$ The question has already been asked, and then deleted, last month: mathoverflow.net/questions/427226/… $\endgroup$
    – Wojowu
    Commented Sep 20, 2022 at 19:54
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    $\begingroup$ I’m voting to close this question because it would be better for HSMSE. $\endgroup$
    – LSpice
    Commented Sep 20, 2022 at 20:12
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    $\begingroup$ Probably derives from Quine's quasi-quotation marks. $\endgroup$
    – Nik Weaver
    Commented Sep 20, 2022 at 20:39
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    $\begingroup$ I voted to reopen, since I find the question interesting on this site. (The fact that a question would also fit on another site is not a reason to close on any one of them, since many questions fit on several sites.) $\endgroup$
    – Joel David Hamkins
    Commented Sep 22, 2022 at 6:36
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    $\begingroup$ also asked at hsm.stackexchange.com/q/14817/1697 $\endgroup$
    – Carlo Beenakker
    Commented Sep 23, 2022 at 5:59

1 Answer 1

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While this is not a full answer, I hope the following observations can still be of some use.

Kreisel and Lévy were using corner quotes, explicitly for Gödel numbers, around the same time as Schoenfield (G. Kreisel and A. Lévy, Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems, Zeitschr. f. math. Logik und Grundlagen d. Math. 14(7-12):97-142, 1968; but note that the paper was submitted on 12 December 1966).

The same notation is used in a related way in three earlier abstracts of Kreisel's (The subformula property and reflection principles, JSL 28(4):305-306, 1963; Reflection principle for Heyting's arithmetic, JSL 28(4):306-307, 1963; Reflection principles and $\omega$-consistency, JSL 28(4):307-8, 1963). There, a distinction is made between free variables $n$ ranging over proofs and $\ulcorner A \urcorner$ ranging over formulae in expressions such as $\mathrm{Prov}(n,\ulcorner A\urcorner)$.

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