> An article by Nic Weaver is worth a read: > > http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1680v1.p > df > > > Here's a quote: > > "An essential incorporation of impredicative > mathematics in basic physics would involve a > revolutionary shift in our understanding of physical > reality of a magnitude which would dwarf the passage > from classical to quantum mechanics [...} the > likelihood of ZFC turning out to be inconsistent [is] > much higher than the likelihood of it turning out to > be essential to basic physics. The assumption that > set-theoretically substantial mathematics is of any > use in > current science is simply false" > > By "impredicative mathematics", he means mathematics > with the powerset axiom. > > I actually think Weaver misses the essential point, > which is this: > > The notion of falsifiability, which is the > cornerstone of science, can be formalized in such a > way that it can be made the cornerstone of > mathematics, and it is eminently reasonable to do so; > if we don't accept falsifiability as part of the > e underlying logic of our mathematics, then our > mathematics is deficient as a language for science. > Impredicative mathematics is not compatible with > falsifiability. > What is your notion of falsifiability other than that of counterexamples which falsify a theory?
> The conclusion is that the claim that it is even > remotely possible that impredicative mathematics > (e.g. ZFC) has an essential role to play in science > is a truly extraordinary claim that requires truly > extraordinary evidence, and such evidence is woefully > lacking. > > ZFC is crackpot mathematics. > > >