Continuum Hypothesis

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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

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