There exists a sset ω that contains all the finite ordinals. By convention, this is the set of nonnegative integers.
Suppose ω is a finite set. Thus a 1-1 function maps all the finite ordinals onto the menbers of the ordinal n, for some finite n. In particular, the first n+1 ordinals map 1-1 into the ordinal n. This contradicts the pigeonhole principle. Therefore ω is not a finite set.
Verify that ω is the union of all finite ordinals, hence ω is an ordinal. If ω is the successor of some set s, then s is an ordinal. In fact s is a finite ordinal, contained in ω. Yet that makes ω the successor of a finite ordinal, whence ω is a finite ordinal, which is a contradiction. Therefore ω is not the successor of anything. It is our first limit ordinal. It is also an infinite ordinal, which makes sense, since it is an infinite set.
remember that the ordinals are well ordered. An infinite ordinal j can't be less than ω. It is either ω, or it contains ω. Infinite ordinals are infinite sets, and ω is the least infinite ordinal.
We can take the successor of ω, similar to ∞+1. Take the successor again to get ∞+2, and so on. The union of all these ∞+n sets is another ordinal, analogous to ∞×2. Take the successor of this to get ∞×2+1, and so on. Who knows where this will lead?
0 = ()
1 = (0)
2 = (0,1)
3 = (0,1,2)
4 = (0,1,2,3)
…
∞ = (0,1,2,3,4,5,6,7,8…)
∞+1 = (0,1,2,3,4,5,6,7,8…∞)
∞+2 = (0,1,2,3,4,5,6,7,8…∞,∞+1)
…
∞×2 = (0,1,2,3,4,5,6,7,8…∞,∞+1,∞+2,∞+3…)
∞×2+1 = (0,1,2,3,4,5,6,7,8…∞,∞+1,∞+2,∞+3…∞×2)
…
∞×∞ = (0+1+2+3…∞,∞+1,∞+2,∞+3…∞×2,∞×2+1,∞×2+2… ∞×3…∞×4 …)
∞×∞+1 = (0+1+2+3…∞,∞+1,∞+2,∞+3…∞×2,∞×2+1,∞×2+2… ∞×3…∞×4 …∞×∞)