Le 06-janv.-06, à 02:40, Benjamin Udell a écrit :
I don't know a whole lot about math, and I tend to be fallibilist, so
I wonder whether anybody really does "know," like Penrose claims, that
those maths are in fact really and truly are consistent, which are
consistent _provably_ only on the unprovable assumption of
arithmetic's consistency. I think of seeming inconsistencies that get
patched up, 0 divided by 0 equals "any number" you want -- so, more
carefully define equality to exclude that problem. Denominators
seemingly turning to 0 in calculus got remedied. And so on.
I think that all mathematicians (99,999...%) believes correctly in the
consistency of Peano Arithmetic (PA). Note that PA is a "simple"
example of a lobian theory or machine).
Few mathematician doubt about the consistency of Zermelo-Fraenkel Set
Theory (ZF), although George Boolos makes a case that ZF could be
inconsistent.
Famous results by Godel and Cohen have shown the relative consistency
of many "doubtful" math assertion: precisely it has been shown that IF
ZF is consistency THEN ZF + the axiom of choice is consistent. The same
for the continuum hypothesis, etc.
Note that (a theorem prover for) ZF is also a Lobian machine.
Much more difficult is the question of the consistency of (rather
exotic) set theories like Quine's New Foundation (NF).
There is no sense to ask about the consistency of the whole math,
because the whole math cannot be presented in a formal theory or
theorem proving machine (if only by Godel incompleteness result). It is
a reason to doubt about Penrose' use of the notion of consistency in
his "godelian" argument against Mechanism/Comp. All what such types of
reasoning show is that: IF I am a sound Lobian Machine THEN I cannot
know which machine I am (and then I cannot know which computational
history extends me, and that is what I use eventually for solving the
OM measure problem).
Bruno
PS I will answer your long "naming-issue" post, Ben, and some others
to, asap.
http://iridia.ulb.ac.be/~marchal/
