limit and infinite ordinals: same thing?

Question

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal.

Is it true the other way around? That is if I have a limit ordinal must that also be an infinite initial ordinal?

This would seem to be true because for a limit ordinal x, if I have any 'y' less than x, then I have a 'w' such that y < w < x.

And the definition of infinite initial ordinal 'x' is: For all B < x, we also have B dominated by (of lesser cardinality than) x. I understand that this means there is no way to have an onto map from B to x. If x is a limit ordinal, then for all y < x, there is no way to make an onto map from y to x. This is true because for whatever supposedly onto function f from B to x you give me, I can always take each image of f (in x), and double that image to produce an ordinal in x which is not mapped to by f.

Does this make sense ?

Thanks in advance for confirming! (or showing false, perhaps with a counter-example?)

score 4 · Accepted Answer · answered Jul 01 '15 at 20:23

4

No. $2\omega$ is a limit ordinal and infinite, but not initial as it is still countable.

answered Jul 01 '15 at 20:23

Hagen von Eitzen

374,180

1

$2\omega=\omega$. (+1 though) – wythagoras Jul 01 '15 at 20:25
Oh yeah.. the countability of these big ordinals can really throw me. Thanks, wythagoras and Hagen for these counter-examples! – Chris Bedford Jul 01 '15 at 20:26

limit and infinite ordinals: same thing?

1 Answers1