# [FOM] The Diagonalization Lemma (historical qn)

Harry Deutsch hdeutsch at ilstu.edu
Fri Jul 6 10:47:14 EDT 2012

```Don't you get the syntactic version from the completeness theorem for T--which I assume is a first order theory?  Harry
On Jul 5, 2012, at 5:06 AM, Peter Smith wrote:

> An historical question, that some FOMer might recall the answer to!
> Distinguish the semantic Diagonalization Equivalence from the syntactic Diagonalization Lemma. Carnap 1934 is often credited with the Lemma. But that's wrong. He gets the Equivalence. Qn: who first explicitly states the Lemma?
>
> To explain. Take the Equivalence to be the claim that given a suitably nice theory T with an interpreted language, and any one-place open T-sentence phi, we can find a T-sentence G such that G <--> phi('G') is true, where 'G' is of course the numeral for the Gödel number of G under some sane coding.
>
> Take the Lemma to be the claim under the same conditions we can find a T-sentence G such that T |- G <--> phi('G').
>
> In Logical Syntax, Carnap gets the Equivalence (and that's what Gödel attributes him in fn. 23 of his 1934 Princeton Lectures). On this basis, Carnap is often/usually credited with the Lemma. But look carefully and it just isn't there. Of course it is a very small step on from the semantic Equivalence to the syntactic Lemma. But it IS a step. So I'm wondering who first explicitly made it.
>
>
> --
> Dr Peter Smith
> http://www.logicmatters.net
>
>
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