continuum hypothesis


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continuum hypothesis

[kən′tin·yü·əm hī‚päth·ə·səs]
(mathematics)
The conjecture that every infinite subset of the real numbers can be put into one-to-one correspondence with either the set of positive integers or the entire set of real numbers.
References in periodicals archive ?
INFINITE ELEGANCE In the decades that followed Cohen's 1963 result, mathematicians trying to settle the continuum hypothesis ran into a roadblock: While some people proposed new axioms indicating the continuum hypothesis was true, others proposed what seemed like equally good axioms indicating the it false, Woodin says.
A good axiom, he felt, should help mathematicians settle not only the continuum hypothesis but also many other questions about Cantor's hierarchy of infinite sets.
However, Woodin suspected a compromise is possible: There might be axioms that answer all questions up to the level of the hierarchy that the continuum hypothesis concerns--the realm of the smallest uncountably infinite sets.
In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved--apart from one missing piece that must still be filled in--that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false.
If there's a simple solution to the continuum hypothesis, it must be that it is false," Woodin says.
Woodin's novel approach of sidestepping the search for the right axiom doesn't conform to the way mathematicians thought the continuum hypothesis would be settled, says Joel Hamkins of the City University of New York and Georgia State University in Atlanta.
Mathematicians haven't yet absorbed the ramifications of Woodin's work fully enough to decide whether it settles the matter of the continuum hypothesis, says Akihiro Kanamori of Boston University (Mass.