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THE CONTINUUM HYPOTHESIS:
There is no set A such that |N| < |A| < |R|
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Requisite definition:
|A| < |B| if there exists an injective function from A to B,
but there does not exist a bijective function from A to B.
(Equivalently, there exists an injective function from A to B,
but not |A| = |B|). As usual, N denotes the natural numbers
and R denotes the reals.
The continuum hypothesis can not, in some sense, be shown true or
false! It was shown by Paul Cohen to be independent of the usual
axioms of set theory.