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 ?
This could be achieved if p([[??].sub.o]) = [[??].sub.1], a formulation of Cantor's Continuum Hypothesis, or generally if p ([[??].sub.[alpha]]) = [[??].sub.[alpha]+1].
If the Continuum Hypothesis holds, then p([[??].sub.o]), the set of all possible subsets of countable, natural, numbers is exhausted by the ordered methods of construction deployed by ordinal generation: p([[??].sub.o]) = [[??].sub.1], or p([omega]) = [[omega].sub.1].
The Continuum Hypothesis states that the formal aspect takes precedence at the infinite level; we can only discern infinite sets that embody some constructible order.
The key example is the proof of the independence of the Continuum Hypothesis, by demonstrating that there is a consistent situation in which this hypothesis fails.
What the above idea suggests is that there IS a minimal relation between ordinal and cardinal number production; it might not be the strict relation of the General Continuum Hypothesis: p([N.sub.[alpha]]) = [N.sub.[alpha]+1].
Often, to pave the way for a proof in B of a contrary of the continuum hypothesis, such as [2.sup.Card]([Omega]) = [N.sub.2], u is chosen so that B [satisfies] u [subset or equal to] [Omega].
Proving the truth or falsehood of Cantor's continuum hypothesis boils down to answering this: Where does the set of real numbers sit in the hierarchy of infinite sets?
Their derision, coupled with Cantor's inability to prove the continuum hypothesis, sent him into several nervous breakdowns.
DECLARATION OF INDEPENDENCE While Cantor was struggling with the continuum hypothesis, other mathematicians were exploring the implications of Cantor's dramatically broad vision of sets.
The continuum hypothesis, however, exposed a glaring incompleteness in these axioms.
Only women were investigated in this study because the continuum hypothesis has only been researched with women.
These results are congruent with previous literature on the continuum hypothesis, which suggests that the fundamental differences among individuals with clinical eating disorders and individuals with milder forms of eating disorders are a matter of degree, not kind (Mintz & Betz, 1988; Nylander, 1971; Rodin et al., 1985; Scarano & Kalodner-Martin, 1994; Tylka & Subich, 1999).