[FOM] Recursion Theory and Goedel's theorems

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Mon Aug 6 07:06:04 EDT 2007

It is certainly true that recursive function theory can greatly illuminate
Godelian phenomena. Indeed, one cannot even formulate general versions of
Godel's theorem without it (as e.g. Kleene has emphasized).

On the other hand, one should keep in mind that recursive function theory
cannot really express things such as consistency, 1-consistency, soundness,
a theory's properly containing another (proving all its theorems and more),
a theory's proving the consistency of another theory etc. - concepts
necessary for proper understanding of Godel's theorems. 

Indeed, from the point of view of recursive function theory, all theories
from the Robinson arithmetic Q to ZFC and beyond have the same degree and
are recursively isomorphic. 

As Kreisel has put it, proof theory begins where recursion theory ends.    

All the Best,


Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
University of Helsinki

E-mail: panu.raatikainen at helsinki.fi


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