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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.