# Continuum Hypothesis

## Hypothesis

There is no set whose cardinality is strictly between that of the integers and the real numbers.

Symbolically, the **Continuum Hypothesis** asserts that $\aleph_1 = \mathfrak c$.

### Independence of ZF and ZFC

In 1940, Kurt Gödel showed that it is impossible to disprove the Continuum Hypothesis (CH for short) in ZF with or without the Axiom of Choice (ZFC).

In 1963, Paul Cohen showed that it is impossible to prove **CH** in ZF or ZFC.

These results together show that CH is independent of both ZF and ZFC.

## Hilbert $23$

This problem is no. $1$ in the Hilbert $23$.

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.6$: Cardinality - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: A set-theoretic approach - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous): Appendix $\text{A}$: Set Theory: Cardinal Numbers