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Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably many? Uncountably many? What are the possibilities to the quantity of counterexamples?

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    I think the answers to [this question](http://math.stackexchange.com/q/191329/8348) answer most of this.2012-11-18
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    As many as you want; see the question to which Arthur linked.2012-11-18
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    The technical answer is "all hell breaks loose" ...2012-11-18
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    @BrianM.Scott can one give an example for such a set assuming that CH is false?2012-11-18
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    @Seyhmus: Sure: $\omega_1$, the set of all countable ordinals.2012-11-18

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