[FOM] The Diagonalization Lemma (historical qn)

Peter Smith ps218 at cam.ac.uk
Thu Jul 5 06:06:01 EDT 2012

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 

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 

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

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