Continuum Hypothesis
Asks whether there is a set size strictly between the integers and the real numbers.
The Continuum Hypothesis, proposed by Georg Cantor in 1878, asks whether 2^aleph_0 = aleph_1 -- that is, whether there exists no cardinality strictly between the countable integers and the uncountable real numbers. It was the first of Hilbert's 23 problems posed in 1900. Godel showed in 1940 that CH is consistent with ZFC (it cannot be disproved), and Paul Cohen showed in 1963 using his revolutionary forcing method that CH is independent of ZFC (it cannot be proved either). Cohen received the Fields Medal in 1966 for this work. The question of whether new axioms should settle CH remains a lively area of research in set theory and the philosophy of mathematics.
Formula
The cardinality of the real numbers (the continuum) equals the first uncountable cardinal -- no intermediate infinity exists.
The generalized continuum hypothesis extends the statement to all infinite cardinals.
CH is independent of ZFC: it can be neither proved nor disproved from the standard axioms of set theory.
Summary
The Continuum Hypothesis, proposed by Georg Cantor in 1878, asks whether 2^aleph_0 = aleph_1 -- that is, whether there exists no cardinality strictly between the countable integers and the uncountable real numbers. It was the first of Hilbert's 23 problems posed in 1900. Godel showed in 1940 that CH is consistent with ZFC (it cannot be disproved), and Paul Cohen showed in 1963 using his revolutionary forcing method that CH is independent of ZFC (it cannot be proved either). Cohen received the Fields Medal in 1966 for this work. The question of whether new axioms should settle CH remains a lively area of research in set theory and the philosophy of mathematics.
