continuum hypothesis

连续统假设

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Other names / variants: CH

Translation evidence: Literature usage

Meaning in context

In ZFC, the assertion $2^{\aleph_0}=\aleph_1$, where $\aleph_1$ is the least uncountable cardinal.

在 ZFC 中断言 2^ℵ₀=ℵ₁，其中 ℵ₁ 是最小不可数基数。

Example

CH rules out a cardinal $\kappa$ with $\aleph_0<\kappa<|\mathbb R|$.

Usage and distinctions

Cantor’s uncountability theorem is provable in ZFC; CH is independent if ZFC is consistent.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Chris Lambie-Hanson — The Continuum Hypothesis, the Axiom of Choice, and Lebesgue MeasurabilityPDF pp.57–63 Real numbers / The Continuum Hypothesis; pp.71–79 Gödel’s Constructible Universe / Forcing · Checked 2026-10-14

Translation sources

1. 复旦大学哲学学院 — 第154期丨杨睿之：追求生活的乐趣，探索逻辑的边界访谈「探索逻辑的边界」中讨论连续统假设的段落（中文名称证据） · Checked 2026-10-14

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