[FOM] Inconsistency of P

Panu Raatikainen panu.raatikainen at helsinki.fi
Sun Oct 2 13:44:58 EDT 2011

Lainaus "Daniel Mehkeri" <dmehkeri at gmail.com>:

> Yes. To quickly clarify: as I understood it, the proof attempt was  
> relying on the fact that if T proves K(n)>m, then T is inconsistent.  
> If a subtheory of T proves K(n)>m, then it does not follow that the  
> subhteory is inconsistent.

I am no sure whether the last sentence is still supposed be a part of  
the proof attempt, or an independent statement of a fact?

(And I must say that I am not at all sure what exactly was the idea of  
the attempted proof - thing started to move much too fast for me in  
page 5 in Outline...)


If c is the constant provided by Chaitin's theorem (for T), then yes,

     If T proves K(n)>c, for any n, then T is inconsistent.

If a subtheory would prove K(n)>c, it is not necessarily inconsistent,  
but then it has to be severely limited theory, and must not be able to  
prove that a Turing machine halts (when that is in fact the case);  
i.e. it must fail to be Sigma_1 complete. That is, it must be more  
limited than e.g. Robinson arithmetic Q.

All the Best


Panu Raatikainen

Ph.D., University Lecturer
Docent in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi


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