What is a Gödel sentence? Is there such a thing as the Gödel sentence of a given theory?

Lajevardi and Salehi, in a short paper from last year, argue against the use of the definite article in the expression ‘the Gödel sentence’, by claiming that any unsound theory has Gödelian sentences with different truth values. We show that the two theorems of their paper are special cases (modulo Löb’s theorem and the first incompleteness theorem) of general observations pertaining to fixed points of any formula, and argue that the false sentences of Lajevardi and Salehi are in fact not Gödel sentences. We conclude with a discussion of the roles played by soundness and truth in drawing further consequences from the incompleteness theorems.

This paper on which this talk is based is joint work with Christian Bennet.