# Is there a set with cardinality greater than N but less than R?

Is there a set with cardinality greater than the natural numbers but less than the real numbers?

Is there a simple proof which shows this, if the answer is no?

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Look up the continuum hypothesis. The short answer is, it's independent of the usual axioms of set theory. – Adrian Petrescu Mar 25 '11 at 6:04
You can also take a look at this question in MathOverflow that has some very interesting answers: mathoverflow.net/questions/23829/… – Apostolos Mar 25 '11 at 7:03

@DJC: No, the axiom of choice is an entirely separate axiom, dealing with the existence of choice functions on arbitrary sets. It is in fact one of the axioms assumed by most mathematicians. The existence of a set with cardinality between $\mathbb{N}$ and $\mathbb{R}$ is indeed referred to as "the continuum hypothesis". – Alex Becker Mar 25 '11 at 6:17