Is it true that $|\mathbb{R}|=2^\omega=\omega_1$?

Question

Note that $\omega_1$ is the successor of $\omega$ and $2^\omega$ is |all functions from $\omega$ to 2|.

Did you see that little box saying "search"? This topic was covered ad nauseum. — Asaf Karagila, Dec 04 '13 at 23:45
Also http://math.stackexchange.com/questions/490411/infinite-sets-and-their-cardinality — Asaf Karagila, Dec 04 '13 at 23:49

score 1 · Answer 1 · answered Dec 04 '13 at 23:43

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This is known as the Continuum hypothesis.

http://en.wikipedia.org/wiki/Continuum_hypothesis.

And this cannot be proved either way using the standard Zermelo–Fraenkel set theory (ZF). Even when the axiom of choice is assumed, it still cannot be proved!

answered Dec 04 '13 at 23:43

user93826

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I find your last remark strange. Why would the axiom of choice have anything to do with proving the continuum hypothesis? If anything, it proves that all the ways of expressing the continuum hypothesis are in fact equivalent. – Asaf Karagila Dec 05 '13 at 00:13
It is probably naivety on my behalf. I personally find it highly non-obvious to whether or not it has an affect. – user93826 Dec 05 '13 at 00:29

score 1 · Answer 2 · edited Apr 13 '17 at 12:21

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The first equality is true. The second is the continuum hypotheses and is independent of the axioms of ZFC. You could also see this question

edited Apr 13 '17 at 12:21

Community

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answered Dec 04 '13 at 23:43

Ross Millikan

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Is it true that $|\mathbb{R}|=2^\omega=\omega_1$?

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