### 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.

## 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, Harry Christen said…

Thanks for sharing such a valuable information with us. Your information increase my knowledge about Continuum Hypothesis and Chaos.

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