Talk:Limit cardinal
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constructing limit cardinals[edit]
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)
- 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)
- 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:
Mistake in the article concerning infinite ordinal omega[edit]
In the article one has , but as it says here, . Why is there the in then? Shouldn't it be after the omega? JMCF125 (discussion • contribs) 15:29, 13 July 2013 (UTC)
- 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)
- Sorry, I hadn't noticed that. Thanks for the clarification. Should I delete this topic off the discussion page? JMCF125 (discussion • contribs) 17:10, 14 July 2013 (UTC)