Carnap and the Diagonalization Lemma

Carnap is often credited with proving the Diagonalization Lemma in Logische Syntax der Sprache. But where does he do it?

Well, in §35 Carnap notes the general recipe for taking a one-place predicate F(x) and constructing a sentence G such that G is true if and only if F(\overline{\ulcorner G \urcorner}) is true, where as usual \overline{\ulcorner G \urcorner} is  the formal numeral for the Gödel number for G.

Fine. But that claim about constructing a semantic equivalence isn’t the Diagonalization Lemma as normally understood, which is a syntactic thesis, not about truth-value equivalence but about provability. It is the claim that, in the setting of the right kind of theory T, then for any one-place predicate F(x) we can construct a sentence G such that T \vdash G \leftrightarrow F(\overline{\ulcorner G \urcorner}). And Carnap doesn’t prove that in §35.

When we turn to §36, where Carnap proves the incompleteness of his system II, again he appeals to his semantic result not the syntactic Diagonalization Lemma. We now take the relevant predicate to be NOT-PROVABLE: so we get a sentence G which is true if and only if it isn’t provable. And then Carnap appeals to the soundness of his system II to argue in the elementary way that G isn’t provable. So at this point Carnap is giving a version of the semantic incompleteness argument sketched in the opening section of Gödel 1931 (the one that appeals to a soundness assumption), and not a version of Gödel’s official syntactic incompleteness argument which appeals to \omega-consistency. Indeed, Carnap doesn’t even mention \omega-consistency in the context of his §36 incompleteness proof. He doesn’t need to.

To summarize so far: in §§35–36, Carnap doesn’t use the theorem that his system II proves G\  \leftrightarrow\ NOT-PROVABLE(\overline{\ulcorner G \urcorner}), i.e. he doesn’t appeal to an application of the Diagonalization Lemma in the modern sense. He doesn’t need it (yet), and he doesn’t prove it (here).

But maybe he returns to the fray later in the book … the story continues!

1 thought on “Carnap and the Diagonalization Lemma”

  1. The proof of the diagonal lemma (as given, for example, in Boolos et al) uses the completeness theorem for first order theories. Specifically, what is here called the “equivalence lemma” is proved valid (i.e. semantically) and then the completeness theorem is applied to obtain the syntactic version. So all Carnap appears to have left out is the application of the completeness theorem. Harry

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