[FOM] Lucas, Penrose, and the Church-Kleene ordinal

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Mon May 31 05:32:48 EDT 2004

"Timothy Y. Chow" <tchow at alum.mit.edu>:

> In other words, I would say that Penrose's argument is best described
> not as trying to use Goedel's theorem to show that human thought is 
> non-recursive, but as a philosophical argument that RP is true and
> hence that "humanly known mathematical truths" are best modeled as
> CK sets.

Certainly there is - and I think there is no dispute on this - a version of 
the Godelian argument against (a quite naive form of) mechanism that is 
sound; roughly, assume someone claims that the mathematical truths he can 
possibly know are captured by certain formal system F he knows with 
mathematical certainty to be sound, or at least consistent. Quite obviously 
this must be false, by Godel's theorems. 

However, the anti-mechanist conclusion of Lucas, Penrose etc. just does not 
follow from this. First, all truths knowable by us may still in fact follow 
from some formal system, but such that we do not know that system, or don't 
know certainly that it is consistent. Or, we may even have some less 
conclusive evidence (short of mathematical certainty) for the consistency 
of the system. etc. 

I say more on these issues in my paper:
"On the Philosophical Relevance of Godel's Incompleteness Theorems",

Those interested in the relation of these issues to transfinite 
progressions of theories etc. might also check:
Shapiro, Stewart (1998) “Incompleteness, mechanism, and optimism”, Bulletin 
of Symbolic Logic 4, 273–302.


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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