Explore: Continuum Hypothesis

In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that 2 ℵ 0 = 2 ℵ 1 {\displaystyle

second smallest is ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly

believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable

integers but less than that of the real numbers, is known as the continuum hypothesis, which has been shown to be unprovable in standard set theories such

in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph

paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an

was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set

{\displaystyle \aleph _{0},\aleph _{1},\dots } ), but unless the generalized continuum hypothesis is true, there are numbers indexed by ℵ {\displaystyle \aleph } that

of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved