# [FOM] The Diagonalization Lemma (historical qn)

Harry Deutsch hdeutsch at ilstu.edu
Mon Jul 9 15:03:27 EDT 2012

```In other words (and this is the last I'll say about this here) the answer to my original question is Yes, not No.  I simply was momentarily doubtful for some reason.  The issue is important in assessing Carnap's contribution.  He did indeed prove the diagonal lemma as we know it, he just may have left out the application of the completeness theorem.  In attributing the diagonal lemma to Carnap, Goedel probably assumed the application of the completeness theorem.  Harry
On Jul 9, 2012, at 9:35 AM, Harry Deutsch wrote:

> I withdraw the question I asked in my last post.  The answer is clearly No.  The question was based on a careless misunderstanding.  Harry
> On Jul 6, 2012, at 9:47 AM, Harry Deutsch wrote:
>
>> 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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