[FOM] Inconsistency of P

[Panu Raatikainen]

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

> I agree with everything you said in this latest message, as long as
> you are assuming T is consistent.

Good!

> But I would like to emphasize that
> your original claim-- that the hypothesized subtheory S must be very
> weak-- is not correct.

Yes - I already took that back, but let us by all means emphasize it.

And thank you for pressing me on this! Thinking it through more  
carefully helped me to clarify certain other issues for myself...

> If S proves "K(n)>c" for some n, then we can
> indeed conclude that T is inconsistent.  But there need not be any
> special deficiency of S in consistency, truth, or strength, since c
> depends on T-- it is c(T), not c(S).

Well, yes, if by c(S) you mean the value given by the Chaitin machine  
construction (though, as I emphazides, S  is then not a subtheory of T)
No, if it denotes the minimal value for which Chaitin's theorem is true.
But I guess we agree here already.

> This was Tao's main point.  If you are still not convinced, please  
> provide some details of the contrary argument you have in mind.

Well, I am not sure if exactly that was his "main point"...

Anyway, though I would insist that Tao and Tausk make several false  
claims, I now think that they are not totally off the rails either.

I came to all this a bit late, and it took some time to digest  
everything, and in particular, the structure of Nelson's proof plan,  
where the crucial part was given in only few lines...

I am now in a process of writing a detailed analysis of what goes  
wrong in the critique of Tao and Tausk, and where they are on the  
right track; and consequently, where exactly the flaw in Nelson's  
proof plan really is. I'll post it asap.


Best

Panu


-- 
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
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/