# 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:

- $\aleph_1 = \mathfrak c$

where:

- $\mathfrak c$ denotes the cardinality of the continuum
- $\aleph_1$ denotes Aleph One.

### Generalized Continuum Hypothesis

The **Generalized Continuum Hypothesis** is the proposition:

Let $x$ and $y$ be infinite sets.

Suppose:

- $\phi_1: x \to y$ is injective

and:

- $\phi_2: y \to \powerset x$ is injective

Then:

- $y \sim x$ or $y \sim \powerset x$

In other words, there are no infinite cardinals between $x$ and $\powerset x$.

## Hilbert $23$

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

## Historical Note

The **Continuum Hypothesis** was originally conjectured by Georg Cantor.

In $1940$, Kurt Gödel showed that it is impossible to disprove the **Continuum Hypothesis** (CH for short) in Zermelo-Fraenkel set theory (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.

Note, however, that these results do not settle **CH** one way or the other, nor do they establish that **CH** is undecidable.

They merely indicate that **CH** cannot be proved within the scope of ZF or ZFC, and that any further progress will depend on further insights on the nature of sets and their cardinality.

It has been suggested that a key factor contributing towards the difficulty in resolving this question may be the fact that Gödel's Incompleteness Theorems prove that there is no possible formal axiomatization of set theory that can represent the entire spread of possible properties that can uniquely specify any possible set.

## 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 - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**continuum hypothesis** - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous): Appendix $\text{A}$: Set Theory: Cardinal Numbers - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**continuum hypothesis** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 5$ The continuum problem