[FOM] inconsistency of P

[Aatu Koskensilta]

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

> By this point it should be clear that I have no problem with "there is a
> proof of `0=1' in PA using at most one million symbols."  As for "PA is
> consistent," I know what it means according to a certain story, and I know
> that those who believe that the story is really true would then conclude
> that, in particular, there is no proof of `0=1' in PA using at most one
> million symbols.  But I myself don't believe that the story is really
> true, and the most concise way to state that in a way that will be widely
> undertood is to say that I don't believe that anyone knows that PA is
> consistent.

   We all naturally have various realist and anti-realist leanings,  
and this line of thought is certainly natural and appealing. If we  
take it to be merely an expression of a certain natural attitude  
there's not much to say, but if we try to take it at face value it  
soon becomes difficult to see just what to make of it. When we say  
that (arbitrarily large) naturals don't really exist, just what are we  
denying? In case of, say, Sherlock Holmes it is clear what is meant by  
saying he didn't really exist (although he had an older brother but no  
sisters), but in case of zero, 2^2^2^2^2^2^500, the first measurable  
cardinal, the situation is different. After all, we don't want to say  
that they might have existed but it just happens they don't. (You  
speak of truth, but it is certainly a trivial consequence of the  
stories we tell about mathematical objects that e.g. arbitrarily large  
naturals exist.)

   Further, if I claim that an algorithm terminates on all inputs in  
polynomial time and explain I know this because Sherlock Holmes and  
Watson have thoroughly investigated the matter, it is a pertinent  
observation that these people don't really exist; but seemingly  
failure to really exist is irrelevant when it comes to considerations  
about naturals, reals, polynomials, large cardinals, etc. It seems  
there is a real difference between Sherlock Holmes existing and our  
merely pretending (for whatever purpose) that they exist. The  
difference between naturals really existing and our just pretending  
they do is more elusive.

   As for Nelson and syntactical matters, naturally he develops (when  
in foundational mode) the relevant mathematics in systems  
interpretable in Robinson arithmetic (by considering systems of  
naturals closed under these and those operations). If we can't rely on  
exponentiation being total, there are subtle issues with coding, but  
mostly it's all standard stuff from the study of weak fragments of  
arithmetic.

-- 
Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus