Generalized Continuum Hypothesis

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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


Sources