Sorry for reviving a two month old thread - I felt like I had to put my own two cents in.

This is just three pages of people repeating what they've been told.

Math is a whole lot more abstract than science, and in that respect people have more freedom to define their own terms. Aside from a few axioms or postulates, all of math is based on logic and reasoning. Furthermore, axioms and postulates (for example 1 + 2 = 2 + 1 or (a + b) + c = a + (b + c)) have their roots in human experience ('If I take two apples and add one apple to it, or if I take one apple and add two apples to it, I get three apples either way.'). To say people are repeating what they've been told is true - but many people (I for one) have understood what I've been told, in the math sense anyway.

Numbers are just abstractions anyway - they represent ideas that help us when we make transactions or evaluate properties of things. So to say 4 is a representation of the set {0, 1, 2, 3} and nothing else is ridiculous. (Who starts counting at zero anyway?) Infinity is a different abstraction, to me meaning "too large in magnitude to imagine." You can use infinity in some equations (like 1/0 = ∞), but once you start trying to evaluate something, infinity becomes invalid. In this sense infinity is not a number.

(As a side note, I do not believe 1/0 = ∞, but the limit as x approaches 0+ of 1/x is infinity. 1/(really small number) = really big number)

I have to say a lot of this argument over whether infinity is a number or not is just semantics. In any cases where mathematicians are trying to use ∞ as a number, it is probably shorthand meaning for "a really really really big number."

[edit] Also, I have always defined an 'infinite set' as a set that is not finite - meaning its cardinality cannot be represented by a whole number (0, 1, 2, ...). Things that are described as infinite can be described analogously.