"But you have to define how it behaves withe respect to the real numbers and the operations defined on the real numbers (infty +infty=infty, 1+infty=infty, infty*infty=infty, and so on and so forth). And the only problems you have are infty-infty, infty/infty, 0*infty and the similar. So you might as well treat it as a number, with these potential problems in mind."
That's part of my problem. If you include Inf with the reals (or integers), they violate the criteria of a ring. 0*Inf must equal 0 for it to be a ring. Nor is it a field, since there is no additive inverse for Inf. It is, however, a semiring. Alas, my background is in engineering math and not pure math, so I'm out of my depth beyond this point. It could be that forming a semiring is sufficient.
Nevertheless, we see at
http://mathworld.wolfram.com/SurrealNumber.html the term "surreal number," which includes the real numbers and the transfinites, and also the term "Omnific Integer." The implication, I think, is that the transfinites are distinct from the reals (or the integers).
I know this part of my comment amounts to argument from authority. Wolfram could be wrong. Or he could be outdated. But if one is going to argue from authority, I reckon we ought to argue from the good one - and this is the best one I could find on the web.
