Continuum Hypothesis and Chaos - smallest-cardinal
I didn't express myself too clearly, sorry about that.
There's a theorem that says that given any set of cardinals, there exists a smallest cardinal which is strictly greater than all cardinals in the set.
Thus, the theorem implies that there must be a least cardinal greater than aleph_0, by taking the set {aleph_0}. Then, one defines "aleph_1" to be this cardinal, whose existence is guaranteed by the theorem.
There's a theorem that says that given any set of cardinals, there exists a smallest cardinal which is strictly greater than all cardinals in the set.
Thus, the theorem implies that there must be a least cardinal greater than aleph_0, by taking the set {aleph_0}. Then, one defines "aleph_1" to be this cardinal, whose existence is guaranteed by the theorem.
posted by So Kung Tower @ 8:40 AM
2 Comments:
At 11:25 AM,
Steve said…
No theorem states that as a group of cardinals, there is the smallest cardinal which is strictly greater than all the cardinals were established.
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At 7:45 AM,
Unknown said…
Thanks for sharing such a valuable information with us. Your information increase my knowledge about Continuum Hypothesis and Chaos.
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