# 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 cardinalities between $x$ and $\powerset x$.

## Notation

The proposition may 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