Quote:
Originally posted by KnifeMissle
Grothendieck, it depends on what you call a number. When most people think about numbers, they think of real numbers, and infinity is certainly not one of those! It's not even a complex number...
I also hesitate to call set cardinalities numbers since I don't know of any operations between them. You seem to be well educated on the subject so perahps you know of some?
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Agreed, the word number has no general definition. I would think that most people think of integers though...
There are three fundamental operations on sets: Disjoint Union (corresponding to addition), Direct Product (corresponding to multiplication) and Power Set (corresponding to exponentiation.
Here's how it goes: Given two sets A and B, say that A <= B if there is an injection from A to B (or equivalently by Cantor-Bernstein, a surjection from B to A). We say that A > B if there is no such injection. We say that A=B if there is a bijection. Note that the use of = is somewhat misleading, if A=B in this sense, then they are not necessarily the same set, they are only equal in cardinality, put otherwise, they are the same cardinal number!
The number A+B is simply the (disjoint) union of A and B. The number AB is the product of A and B, and A^B is formally defined as the set of functions from B to A. In particular, 2^B is the set of functions from B to a two-element set, and we can identify 2^B with the set of subsets of B (check this!).
Now we can represent any positive integer n by a set with n elements (e.g. 0 is the empty set), and arithmetic with positive integers is part of the above mentioned cardinal arithmetic.
Now for some fun:
if A is infinite, and B is any set, then A+B=max(A,B). For example, if A is infinite and B is finite, then A+B=A.
We always have A<2^A by an adaption of Cantors diagonal trick.
0 times A is always 0.
What you can see is that we don't get a group or ring, but we do get some operations.
Here's some food for thought: Given an infinite set A, is there a set B such that A < B < 2^A (with strict inequalities?). The continuum hypothesis says that for A=(all natural numbers) there is no such set B. By the way, we can prove that 2^A = (all real numbers). For general infinite A, this is what is called the generalized continuum hypothesis (GCH). It turns out that this question is independent of the usual set theoretical axioms we use to construct mathematics.