[FOM] Re: On Gödel's Enigmatic Footnote 48a

[praatika@mappi.helsinki.fi]

Dear Jeff

You wrote:

> What bothers me is Gödel's claim:
> 
>       Higher-Order Decidability Claim:
>       "The undecidable propositions here presented become always
>       become decidable by the adjunction of suitable higher types."
:
> But how could Gödel have known about this sort of phenomenon even in
> 1931? He doesn't give the slightest indication how the proof goes. 

Maybe he just thought that in a strong enough set theory or type theory, 
one can prove the existence of a model for the original theory. (remember 
that he had already proved completeness theorem, in terms of intuitive 
notion of truth.) - so I remain sceptical...

BTW, Feferman dicusses this and related passages in his 
"Gödel's program for new axioms: Why, where, how and what?,"
see his homepages. 

> Polish book). In the Postscript, Tarski wrote,
> 
>          The definition of truth allows the consistency of a deductive
>          science to be proved on the basis of a metatheory which is
>          of higher order than the theory itself. 

Which is, as I always like to remind, false. ACA_0 can give an adequate 
truth definition for PA, but can't prove Cons(PA)...  

Best

Panu




Panu Raatikainen

PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
 
Address:
Helsinki Collegium for Advanced Studies
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E-mail: panu.raatikainen at helsinki.fi
 
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm