V.Sazonov at csc.liv.ac.uk
Fri May 7 09:37:28 EDT 2004
I also found that your proof does not work and have no idea how
it could be corrected.
Saying that, I also disagree with the name of your file disaster.ps
and want to answer to your another question (although probably
in a different manner than your own comments on that):
> Can one form an intuitive picture of what the world would be like if
> PA were not consistent?
First, what is the "world"? Second, inconsistency of PA or ZF would
not be a full disaster for mathematics but
(i) a good lesson for all of us, especially for platonists (if they
still continue to be platonists);
(ii) a lot of very interesting work on reconsidering foundations of
most essential parts of existing mathematics and theoretical computer
(iii) the role of foundations of mathematics for the whole mathematics
and from the point of view of all mathematicians would become
(iv) all "normal" mathematicians would continue to work as before
hoping that some (sufficiently reliable) foundations for their work
will eventually be found by people working in foundations;
(v) all "normal" mathematicians would someway (may be very strongly)
change their style of work when new foundations will be suggested
(like it was in the previous century when set theory became the
(vi) reconsidering applications of mathematics in physics, computer
science, etc. with probably highly interesting consequences;
(vii) new foundations may be also found to be contradictory.
GO TO START
END OF FATASY
By the way, the contemporary role of foundations (such as logic,
lambda calculus, finite model theory, etc.) seems much higher in
TCS than in the traditional branches of Mathematics.
Even once a mistake have been found, the very possibility to think
that there is no genuine guarantee that PA or ZF is consistent
(we have only a hope based on an intuition which is, in principle,
unreliable so that our hopes may be suddenly destroyed) is an
occasion to start thinking on foundations at least little bit
P.S. By this posting I do not intend to start another thread in FOM,
at least because I have not enough time for participating and,
unfortunately, cannot guarantee my answering.
More information about the FOM