Cardinality #2
|N| < |R| (N= set of natural numbers, R= set of real numbers, | ... | = Cardinal of ...). Does there exist a number between |N| and |R|,i.e, |N| < < | R |?
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1 solution
It 's called the continuum hypothesis. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in the year 1900. Τhe answer to this problem is independent of ZFC set theory (that is, Zermelo–Fraenkel set theory with the axiom of choice included), so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.