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by sykh 2926 days ago
The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.
2 comments
jordigh 2926 days ago
The continuum hypothesis being independent just means that it's an additional rule you can add or remove from the game you are trying to play. It doesn't mean we are lacking in knowledge and that if we were to work harder we would solve this problem. We do know which aleph c is: it's aleph_1 with CH and some other aleph without CH. Just take your pick which version you like better.

It's not like we don't know which one is the true model of military combat: chess or checkers. They're just two different games with two different rule sets, and you get to pick which one you like to play more.

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sykh 2925 days ago
The set theory that most working mathematicians deal with is ZFC. In ZFC it is not known what cardinal the continuum is. Hence the statement that I was responding to is incorrect. The person I responded to said that they do know how many reals there are.

The cardinality of the reals is called c. It is known to the be the same as the cardinality of the power set of the naturals. It is not known, in ZFC, which aleph this is. We just know that it is the same as the size of another set.

If you want to add an axiom and say that c is aleph1 then you are free to do so. But if you don't have this axiom then you don't know which aleph it is. So in what sense can you say that you know how many reals there are? You only know it if you add an axiom that says, "It is aleph1."

If I have a jar of pennies and I know it has the same number of pennies as the number of quarters in another jar that I have, does this mean I know how many pennies are in the jar?

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OskarS 2926 days ago
Exactly right.
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throwaway080383 2926 days ago
I think the issue is with the implicit claim that we don't "know" a cardinal until we know which aleph it corresponds to.
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