[FOM] inconsistency of P
Aatu Koskensilta
Aatu.Koskensilta at uta.fi
Mon Oct 3 15:57:53 EDT 2011
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
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