In order to help me understand my infinites, I started reading up on Ordinal Numbers on MathWorld. I’ve run into three snags so far. If you know the answers to these, feel free to be terse; I don’t need an in-depth explanation.

My snag has to do with A and B being disjoint. If 2 = {0, 1} and 3 = {0, 1, 2} then 2 and 3 clearly aren’t disjoint. Is 2 + 3 undefined?

[Quote]

Let * (A, <=)* and

*be totally ordered sets. Let*

**(B, <=)***be the Cartesian Product and define order as follows. For any*

**C = A × B***and*

**a[sub]1[/sub], a[sub]2[/sub] in A***,[ol][li]If*

**b[sub]1[/sub], b[sub]2[/sub] in B***, then*

**a[sub]1[/sub] < a[sub]2[/sub]***,If*

**(a[sub]1[/sub], b[sub]1[/sub]) < (a[sub]2[/sub], b[sub]2[/sub])***, then*

**a[sub]1[/sub] = a[sub]2[/sub]***and*

**(a[sub]1[/sub], b[sub]1[/sub])***compare the same way as*

**(a[sub]2[/sub], b[sub]2[/sub])***(i.e., lexicographical order)[/ol][/li][/Quote]*

**b[sub]1[/sub], b[sub]2[/sub]**This seems straightforward enough, but I get the opposite result as I expect from transfinite multiplication. That is:

omega × 2 = {(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1), ··· } = omega

2 × omega = {(0, 0), (0, 1), (0, 2), (0, 3), ···, (1, 0), (1, 1), (1, 2), ···} = omega + omega

What’s going on here?

- Ordinal Exponentiation. This is the one that’s really bugging me.

I don’t even understand what this is saying. Is * supposed to denote ordinal multiplication? What does gamma have to do with anything?