[FOM] The Diagonalization Lemma (historical qn)
Harry Deutsch
hdeutsch at ilstu.edu
Mon Jul 9 14:03:13 EDT 2012
The completeness theorem is used in the proof of the diagonal lemma--at least it's used in the proof given in Boolos, Burgess, and Jeffrey. So I wasn't so far off after all. I now don't really understand Peter Smith's question. 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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