rationalskepticism.org

surreptitious57 wrote:But you cannot add anything onto infinity because it is never ending.

Oh, you totally can. This is what the ordinals are all about. "Ordinal" isn't a fancy maths word. It just refers to a use of numbers to order things: first rather than one. Second rather than two. Third rather than three. One hundredth rather than one hundred.

It's perfectly acceptable to have infinitieth, and, more, the idea of infinitieth is almost demanded when one starts thinking seriously about ordering. The infinitieth number is that object that you append to the end of the counting numbers, normally written ω. It is defined as an extra object such that all finite numbers are stipulated to be smaller than it.

With ω, you could imagine a decimal number where there are an infinite number of digits after the decimal point, plus one extra at the ω place.

Naturally, you don't have to stop at ω. There is ω + 1, defined as an extra object such that all finite numbers and ω are stipulated to be smaller than it.

Of course, there is ω + 2, ω + 3, and so on, and when you hit the limit there, you introduce ω + ω. This game goes on for a long time, even if you only introduce a countable number of these new objects. Then things get weird.

There's an axiom you can assert which says that the real numbers can be enumerated by ordinals. There, you will have a first real, a second real, a third real, and so on, up to the ωth real, and then the (ω+1) real, and so on, up to the (ω + ω) real, and so on, up to the ordinal 2^ω (two to the power of ω). The ordering implied here isn't the usual ordering of the reals, of course.