Who's afraid of Kurt Gödel?

If I'm reading what you guys are saying, you're talking about Gödel not applying to mathematical systems with a finite set of numbers. In other words, in a mathematical space where there was a 'largest number', Gödel doesn’t apply (because you can prove anything simply by testing your theory on every number). That's cute, but it seems like a rather dull universe (although somewhat practical, because in the real world we are always working with finite amounts of stuff. Hmm.) But Andrew, I definitely recommend you click around on Wikipedia and learn more about the mathematics at the time (like Hilbert's program, which I just found out about today) and how Godel's proof really demolished the whole thing.

this thread continues at http://www.acooke.org/cute/MoreConstr0.html

and, finally(?), that faq includes a passing reference to constructive zf set theory which, presumably, is zf without the axiom of infinity (and without the axiom of choice, if i understand the relationship between that and the excluded middle, which i probably don't, but never mind). so it looks like the connection this approach has with computers (that proofs are effectively algorithms) has led to some kind of revival in its study. which is nice.

and a nice http://www.math.canterbury.ac.nz/php/groups/cm/faq/

and http://plato.stanford.edu/entries/mathematics-constructive/ gives a pretty good description of constructivist logic (and its limitations). it also shows the relationship to computing. (and at one point there's a derivation of the axiom of choice (in a particular variety of modern constructive mathematics), but by that far down the page i was pretty much lost...)

incidentally, there's an example of the constructivist approach to real analysis in the http://en.wikipedia.org/wiki/Mathematical_constructivism on constructivism.

Hi Andrew. Foundations of mathematics really isn't my area, so I'll be the first to admit that what I say should be taken with a grain of salt. But, to be fair, I never said that calculus would be impossible, just calculus as we know it. I can imagine some discrete version of it. Also, measure theory is only one ingredient of calculus. In the end, I haven't encountered any respected mathematician who seriously questioned infinity. In fact, even the ones who just question choice are kind of looked at as lunatics.

heh. the quote comes from dewitt (the famous physicist). it makes no sense for a physicist to be pushing the idea that calculus is impossible. and he gives as an example measure theory (integration)...

hi epimorph - what i'm interested in is how complete it's possible to be without the axiom of infinity. i've just been skimming through my basic intro to set theory over dinner and while they spend some time discussing the axiom of choice, they don't bother to question infinity. yet the quote says "all of modern mathematics". if that's the case then it seems quite odd, to me, that "everyone" is so happy to include it. it just struck me as odd that there's that review - apparently written by a sane and reasonable mathematician - that happily discards an axiom of mathematics that (almost) everyone assumes is completely necessary. and that he can get away with it and still produce, apparently, a large fraction of what we have. don't you think that's odd? and what about those other "necessary" axioms? as for your final point - i'm more interested in what he's using for a definition of "interesting" rather than trying to find holes in what he says. after all, presumably he's aware of the continuum hypothesis. in other words: the fact that he writes that suggests that there is a definition of "interesting" for which it makes sense, and i'm curious what that might be. (on preview - cool, thanks. that's exactly what i was looking for. although i'm not sure you're right. it's a pretty sketchy definition of "all of modern maths" that excludes real numbers. so i half suspect (no offence!) that there's some alternate approach, although i haven't a clue what it is.)