zlacker

>>jeremy+(OP)
Mathematician Edward Nelson has a very interesting take on "big" numbers. He claims that the exponential function is not necessarily total, and a number like 2^1000000000000 might not actually exist. The reason he singles out exponentiation is that the reasoning that exponential numbers are "real" numbers, is circular (impredicative). According to Nelson, speaking about such numbers might lead to condradictions, just like speaking about "the set of all sets that don't contain themselves" leads to a contradiction.

He also relates these issues to Christian philosophy, which I find very interesting. In particular, he claims that the a priori belief in the objects defined by Peano Arithmetic, is equivalent to worshipping numbers, as the Pythagoreans did.

I think this is the best starting point if you're interested in reading about his ideas: https://web.math.princeton.edu/~nelson/papers/warn.pdf

>>swatow+im
I'll read the paper, it sounds interersting. But isn't all Math "circular" -- Goedels Second Incompleteness Theorem yeah?

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...

>>hellba+Nm
Goedel's incompleteness theorem uses Peano Arithmetic, which Nelson claims is (possibly) inconsistent. He develops his own "predicative" arithmetic, to which Goedel's theorem does not apply.

(large pdf) https://web.math.princeton.edu/~nelson/books/pa.pdf

>>swatow+Rm
I'll read it, thanks. I always (layman) understood Goedel's theorem as meaning you can be complete or consistent but not both. So you can have a language that can describe everything but you'll have paradoxes, or you'll have no paradoxes but not be able to describe everything (all possibilities) in your system

>>hellba+4p
Yes, Goedel's Theorem says you can be complete or consistent but not both, but only about systems that are stronger than Peano Arithmetic, i.e. systems that contain the axioms of Peano Arithmetic, as well as any other axioms.

Usually this is left out, because Peano Arithmetic is treated as a MVP for mathematics. But Nelson claims Peano Arithmetic may be inconsistent, and proposes a weaker system.

>>swatow+Hp
Wow. I have a lot of reading to do. Thanks!