English Dictionary

continuum hypothesis

名詞

(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.

日本語の意味

連続体仮説は、実数全体ℝの無限部分集合が、必ず自然数全体の濃度（可算無限）か、あるいは実数全体の濃度（非可算無限）のいずれかに等しいという仮説である。

Many set theorists continue to debate whether the continuum hypothesis can be resolved within ZFC or requires additional axioms.

多くの集合論者は、実数の任意の無限部分集合が可算か実数全体と同じ濃度を持つとする連続体仮説がZFC内で決定できるのか、あるいは追加の公理が必要なのかを議論し続けている。

意味（1）

(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.

In other words, it is consistent with the axioms to suppose that the continuum hypothesis is false. (It had been known for some time — see below — that it is consistent with the axioms of set theory to suppose that CH is true.)

意味（2）

(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.

Under the continuum hypothesis, the material being studied, whether it be solid liquid or gas, is assumed to be continuous.