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](https://plato.stanford.edu/entries/goedel-incompleteness/).


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.



First use of corner quotes for Gödel numbers
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](https://doi.org/10.1002/malq.19680140702), 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](https://www.jstor.org/stable/2271325) (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 $\mathrm{Prov}(n,\ulcorner A\urcorner)$.


