[FOM] Proof "from the book"

[praatika@mappi.helsinki.fi]

Jeffrey Ketland wrote:

> Somewhere in Gödel's recollections (either in the Wang volumes, or in
> the Collected Works, or in Feferman's recent book), he says that, in his
> train of thought in 1930, he found the Indefinability Theorem first, and 
> then the first Incompleteness Theorem came slightly later. I can't 
> remember exactly where this is located. If I remember right, the gist is 
> this. 

There is a rather nice discussion of this issue in:
Roman Murawski: Undefinability of truth. The problem of the priority: 
Tarski vs. Gödel, History and Philosophy of Logic 19 (1998), 153-160.
one can find the paper in Murawski's homepage:
http://www.staff.amu.edu.pl/~rmur/


> While this is well-known, I am tempted to speculate that a lot of the 
> technical material that appeared in Tarski's Der Wahrheitsbegriff had 
> already been discovered by Gödel himself,
:
> seems to me that Gödel must have already worked out the details of
> formalized semantics before his 1931 paper appeared. If this is right,
> then Gödel must have anticipated quite a few of the ideas we usually
> associate with Tarski's 1933 work (the T-scheme, the inductive truth 
> definition, the definability of nth-order truth at (n+1)th-order, 

I don't think this is true. Gödel was working, like pioneers of model 
theory such as Löwenheim and Skolem, with an intuitive notion of "truth in 
a model". 

I think Gödel's version of the undefinability theorem was in terms of 
representability: truth can't be represented in, say, PA. (If I remember 
correctly, this what Gödel does in his 1934 lectures.)

I would argue that Tarski's version is quite different from that; moreover, 
contrary to the popular view, it does not use semantical concepts, but is 
purely syntactic and very general. Tarski's ingenious and original idea was 
to use T-sentences: roughly, a theory cannot have in its language a 
predicate T such that it proves all instances of T-sentences 
(i.e. T([s]) <-> s ]
(Gödel did no such thing.) 
This has been argued convincingly in a forthcoming paper by Mario Gomez-
Torrente. 

Obviously, I thus also disagree with Avron:
>(Historically, Tarski's proof was a reproduction of Godel's
>proof, not the other way around...).


All the Best

Panu


Panu Raatikainen

PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
 
Address:
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E-mail: panu.raatikainen at helsinki.fi
 
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