The article only shows it is not an ordinal number.
Because I'm a Wikipedian, I've learned that whenever I visit an article on Wikipedia, Wikibooks, etc. I can visit the article talk page too. Many of the usual misconceptions about infinity can be found in the talk page of that article. Another article showing that infinity is not a number
by a young mathematician with some demonstrated chops in mathematics.
The last discussion of this issue on HN, which appears to have been from about two years ago, was interesting, so when I saw the article submitted here today (while looking up sources for the teaching I do), I thought I'd invite HN participants to discuss the issue again.
Two follow-up questions:
1) What do you mean by number?
2) Supposing the claim is that infinity is a number, how would that claim be verified by accepted principles of mathematics?
I think it is fair to say that a number should be an object of a ring. I'm an algebraist though. That is, there should be an addition operation and a multiplication operation that satisfy certain conditions.
1. Calculus. One sees in calculus things like
Lim x->5 f(x) = infinity
In this case infinity is not a number but rather a shorthand notation for a more complicated statement. What is meant by the use of the infinity symbol is that the limit is not bounded in the real number system. More specifically that given any large real number I can find a number d such that whenever |x - 5| < d then f(x) > M. Here the use of infinity is not meant to be as a number though making such an association is helpful to beginners in terms of visualizing what is going on.
2. There are ordinal numbers that are infinite. That is, that represent the order type of an infinite well ordered set. Ordinal numbers do not for a ring but they do have an arithmetic defined. They do form a semi-ring though. If one wants to say that objects of a semi-ring are numbers then there are infinite numbers. This also applies to cardinal numbers.
3. There is the extended real number system which has the symbols -infinity and positive infinity attached to the real number system to form a compact set. Think of the compact closure of the reals. Again not a ring though.
4. That said, there are infinite sets. An infinite set is one that is not finite. Or, in more precise terms, and infinite set is one that can be put into 1-1 correspondence with a proper subset of itself. Cardinal numbers and cardinal number arithmetic deal with comparing sizes of sets. Mostly useful for dealing with infinite sets. Some infinite sets are bigger, in a meaningful way, than other infinite sets. The set of reals is much bigger than the set of integers.
A while back we had a discussion about the halting problem, and someone had difficulty with the existence of the paradox in the canonical proof: http://news.ycombinator.com/item?id=1323407
The difficulty was, I think, in realizing that if we make a bunch of claims, and those claims lead to a paradox, the paradox exists, and that is not allowed. We can't "work around" the paradox, because paradoxes are not allowed to exist. Nor can we conveniently try to redefine some of our constructs such that they don't encounter the paradox if those constructs live in a reality with constructs that do encounter the paradox. I think it's a similar mental hurdle that some other people have with understanding that infinity is not a number.
