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Is there a name for this ordinal $\omega$, just like its cardinal $\aleph_0$ has a name "aleph null set"?

Whenever I do my assignment about cardinality of sets (not in set theory class), if I want to make my proof precise, I introduce $\omega$ as "the least infinite ordinal" or "the least limit ordinal". I wonder if there is a fancy word like "aleph-null set".

Kamil Jarosz
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John. p
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    It's mostly called "omega" in my experience. If you're talking about ordinals, and you say "omega", then I am almost certain there won't be any confusion. – Arthur Mar 31 '14 at 09:28
  • @Arthur If there is a confusion, then is there any other name substitute to omega? – John. p Mar 31 '14 at 09:31
  • @Arthur If it is my own work, it's fine. However, i'm asking this since there are few professors don't care about precise arguments about cardinality but hand out assignments proving some statements about cardinality. – John. p Mar 31 '14 at 09:36
  • In that case you start by saying "Let $\omega$ be the first infinite ordinal", and from then on you can continue without worry. – Arthur Mar 31 '14 at 09:41
  • How about omega-naught? (Since sometimes $\omega_0$ is used to denote $\omega$.) – Burak Jan 03 '16 at 12:07

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