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Countable Ordinals

Post Icon Posted: Submitted by cosmin on 27 November 2007 - 4:15am.

Joined: 2006-11-02
Posts: 1019

Turns out you were actually quite right about that Tom ($ \omega^\omega $ and even $ \epsilon_0 $ are countable), but it's not that bad since $ 2^\omega $ is also countable, and so are pretty much all ordinals which we can express in terms of $ \omega $. It's not actually much of a problem, since cardinal exponentiation and ordinal exponentiation are not the same thing at all ($ 2^{\aleph_0} = c $ but $ 2^\omega = \omega $), so the correspondences we were making between them aren't valid.

http://en.wikipedia.org/wiki/Ordinal_arithmetic

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Post Icon Posted: 27 November 2007 - 2:33pm

Joined: 2007-10-03
Posts: 383

Yeah that's actually the problem, you can identify cardinals with ordinals, but the difference in notation is just there to differentiate between the arithmetic of the two (ordinal/cardinal exponentiation).
Quite confusing indeed. Anyhow, thanks for clarifying that up.
Post Icon Posted: 29 November 2007 - 4:57pm

Joined: 2006-10-01
Posts: 427

Hate to say I told you so.

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