continuum hypothesis

Noun

Meaning

(set theory) The hypothesis which states that any infinite subset of ℝ must have the cardinality of either the set of natural numbers or of ℝ itself.

(physics, kinematics, continuum mechanics) The assumption, for the purposes of mathematical modelling, that the material being studied is a continuous mass rather than being composed of discrete particles.

Synonyms

Examples

Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis ) imply the axiom of choice and hence a well order of the reals. Source: Internet

Forcing main Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Source: Internet

Examples of these axioms include the combination of Martin's axiom and the Open colouring axiom which, for example, prove that (N*) 2 ≠ N*, while the continuum hypothesis implies the opposite. Source: Internet

At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. Source: Internet

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. Source: Internet

Furthermore, using techniques of forcing ( Cohen ) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Source: Internet

Close letter words and terms