[FOM] Proof "from the book"

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Tue Aug 31 04:57:14 EDT 2004

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:

> 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-

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 Raatikainen

PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi

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