constructing limit cardinals

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The article says

An obvious way to construct more limit cardinals of both strengths is via the union operation:   is a limit cardinal, defined as the union of all the alephs before it; and in general   for any limit ordinal λ is a limit cardinal.

Is that obvious? Is it even a theorem of ZFC? I mean, how do we even know that  ? The continuum hypothesis article doesn't say anything about this. Maybe I'm just reading the sentence incorrectly? If not, the article could use some added clarification. Thanks 66.127.52.47 (talk) 03:03, 14 March 2010 (UTC)Reply

The text you quoted does not say  , it just says that   is a limit cardinal. In general if an ordinal   is the limit (union) of a sequence   then   is the limit of  . — Carl (CBM · talk) 11:48, 14 March 2010 (UTC)Reply
I see what you mean now. I added "weak" to clarify that   is a only weak limit. Here is what we know about the relationship between   and   in ZFC:
  • ZFC proves that   is not equal to  , because of König's theorem (set theory)
  • ZFC does not prove either   or  . The best intuitive way of understanding Cohen's result is that   can be any uncountable cardinal with uncountable cofinality.
— Carl (CBM · talk) 12:00, 14 March 2010 (UTC)Reply

Mistake in the article concerning infinite ordinal omega

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In the article one has  , but as it says here,  . Why is there the   in   then? Shouldn't it be after the omega? JMCF125 (discussioncontribs) 15:29, 13 July 2013 (UTC)Reply

See Ordinal arithmetic. a+ω=ω only holds when a is less than ω, that is, when a is finite. The α to which this article refers is intended to be any ordinal including infinite ordinals. Assuming the axiom of choice, the article is correct in saying that   is a strong limit ordinal for any ordinal α. JRSpriggs (talk) 07:39, 14 July 2013 (UTC)Reply
Sorry, I hadn't noticed that. Thanks for the clarification. Should I delete this topic off the discussion page? JMCF125 (discussioncontribs) 17:10, 14 July 2013 (UTC)Reply
Usually we leave discussions which are relevant to the article even if they have been concluded. Someone else may have the same concern that you had and be enlightened by this discussion, or choose to revive it. JRSpriggs (talk) 20:45, 14 July 2013 (UTC)Reply