Generalized Continuum Hypothesis

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 $\powerset x$.

Notation

The Generalized Continuum Hypothesis can be abbreviated $\operatorname {GCH}$.

Historical Note

The Generalized Continuum Hypothesis was originally conjectured by Georg Cantor.

It is a straightforward generalization of the Continuum Hypothesis.

Sources

• 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