[FOM] Inconsistency of P

Panu Raatikainen panu.raatikainen at helsinki.fi
Mon Oct 3 02:47:08 EDT 2011

Lainaus "Monroe Eskew" <meskew at math.uci.edu>:

> I was not claiming "K(n)<c" is true in my example.

I guess I was too quick.

In any case, the essential point is that for all Sigma_1 complete  
theories, (simple) consistency equals to Pi_1 soundness, i.e. that the  
theory does not prove any false Pi_1 sentences.

If, as in our original example, a theory T is inconsistent, because it  
proves  "K(n)>c" for some n, it must be that it proves *some* false  
Pi_1 sentence F as a consequence.

So if a subtheory S also proves "K(n)>c" but is consistent, it must  
either be Sigma_1 incomplete, or somehow block the derivation of F  
from  "K(n)>c".


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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