[FOM] Provability of Consistency
Anton Freund
freund at mathematik.tu-darmstadt.de
Mon Apr 1 04:10:02 EDT 2019
Dear Sergei,
Thank you again for your explanations! I agree that we should not prolong
this line of discussion much more, because I think that most arguments
have been made explicit. Nevertheless I would like to clarify my own
position with respect to our last exchange:
I do not at all intend to question any standard notions of logic, such as
the first-order language or the Peano axioms. Note that my worries about
the Peano axioms did not arise from my own position, but from an extreme
case that I devised to understand yours. My own conclusion from this
extreme case would be that your use of standard and non-standard numbers
is not suitable to analyse the question at hand.
It is probably fair if I commit to something positive and sketch my view
on consistency: I think that there are several possible approaches, but in
the end I would probably settle for the one described by Timothy Chow [FOM
post from Fri Mar 29 12:15:05]: Mathematical experience suggests that any
finitistic proof of a statement about finite and certain infinite objects
(natural numbers, syntax, finite graphs, computable sets ...) can be
mimicked by a formal PA-proof of the natural formalization of this
statement (without any domain conditions). As Tim points out, if we accept
this claim and Gödel's theorem, then we must accept that the consistency
of PA cannot be proved by finitary means (whatever our view on
non-standard numbers).
To conclude I would like to comment on your examples of reals vs. complex
numbers and integers vs. natural numbers. I do not find these examples
convincing, because I do not think that the situation is analogous to
standard vs. non-standard. In any suitable mathematical context there will
be a well-determined set of reals and a well-determined set of complex
numbers. But there is no set of non-standard numbers – the notion of
non-standard number does only make sense with respect to a specific model.
Similarly, there is no set of standard numbers. There is just a set of
natural numbers (and, of course, a set of integers, of reals, of complex
numbers ...). I think that this difference to the case of reals vs.
complex numbers is important.
I would also like to thank you for the discussion!
Best,
Anton
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