> One other thing, just as an aside, "does infinity exist as a real thing?" I know
> it exists as a concept, but is anything infinite in scope?
There is an ongoing debate on this among mathematicians. The "Intuitionists"
notably Brouwer, argue that it is not valid to use the Law of the Excluded Middle.
That is, just because A is not true, you can't conclude that "not A" is true.
For example, consider the following.
Theorem: There are infinitely many prime numbers.
Proof: Suppose there are only finitely many primes. Let N be the number of primes.
Now we know that there are some primes, so N is a positive integer. We can list
the primes: p1, p2, p3, ... ,pN. Now let Q = p1*p2*p3*...*pN + 1. Then Q
is not divisible by any of p1, p2, ... ,pN, so Q is another prime, bigger than all
the p1, p2, ... ,pN, which contradicts the assumption that the list p1, p2, ... ,pN
contains all the primes. //
For the Intuitionists, there are no infinite sets. Most of calculus is impossible.
The vast majority of mathematicians choose to adopt the Axiom of Infinity
which states that there does exist an infinite set. Of course it is just an assumption,
but it gives a much more interesting theory, to me at least.
Another seemingly innocuous axiom is the Axiom of Choice, which basically says
that, given any collection of sets, it is possible to choose an element out of each set.
This seems harmless enough, and in fact most of mathematics is gone if you don't
accept it, however it leads to some paradoxes, like the Banach-Tarski Paradox:
a way of separating a solid sphere into 5 congruent pieces, then reassembling them
into a solid sphere with twice the radius!
"If the doors of perception were cleansed, everything would appear to man
as it is, infinite." -William Blake
Bill Haloupek
haloupekb@uwstout.edu
http://www.mscs.uwstout.edu/~billh/home.html