# Continuum equals Cardinality of Power Set of Naturals

## Theorem

$\mathfrak c = \card {\powerset \N}$

where

- $\powerset \N$ denotes the power set of $\N$
- $\card {\powerset \N}$ denotes the cardinality of $\powerset \N$
- $\mathfrak c = \card \R$ denotes the continuum.

## Proof

By Reals are Isomorphic to Dedekind Cuts there exists bijection:

- $f: \R \to \mathscr D$

where:

- $\mathscr D$ denotes the set of all Dedekind cuts of $\struct {\Q, \le}$.

Dedekind's cuts are subsets of $\Q$.

Therefore by definition of power set:

- $\mathscr D \subseteq \powerset \Q$

By Subset implies Cardinal Inequality:

- $\card {\mathscr D} \le \card {\powerset \Q}$

By Rational Numbers are Countably Infinite:

- $\Q$ is countably infinite.

Then by definition of countably infinite there exists a bijection:

- $g: \Q \to \N$

By definition of set equivalence:

- $\Q \sim \N$

Hence by definition of cardinality:

- $\card \Q = \card \N$

Then by Cardinality of Power Set is Invariant:

- $\card {\powerset \Q} = \card {\powerset \N}$

By definition of set equivalence:

- $\R \sim \mathscr D$

Hence by definition of cardinality:

- $\card \R = \card {\mathscr D}$

Thus:

- $\mathfrak c \le \card {\powerset \N}$

Define a mapping $h: \map {\operatorname {Fin} } \N \times \powerset \N \to \R^+$:

- $\forall F \in \map {\operatorname {Fin} } \N, A \in \powerset \N: \map h {F, A} = \displaystyle \sum_{i \mathop \in F} 2^i + \sum_{i \mathop \in A} \paren {\frac 1 2}^i$

where $\map {\operatorname {Fin} } \N$ denotes the set of all finite subsets of $\N$.

A pair $\tuple {F, A}$ corresponds to binary denotation of a real number $\map h {F, A}$.

It means that $h$ is a surjection.

By Surjection iff Cardinal Inequality:

- $\card {\map {\operatorname {Fin} } \N \times \powerset \N} \le \card {\R^+}$

By definition of subset:

- $\map {\operatorname {Fin} } \N \subseteq \powerset \N$

Then by Subset implies Cardinal Inequality:

- $\card {\map {\operatorname {Fin} } \N} \le \card {\powerset \N}$

\(\ds \card {\map {\operatorname {Fin} } \N \times \powerset \N}\) | \(=\) | \(\ds \max \set {\card {\map {\operatorname {Fin} } \N}, \card {\powerset \N} }\) | Cardinal Product Equal to Maximum | |||||||||||

\(\ds \) | \(=\) | \(\ds \card {\powerset \N}\) |

Because $\R^+ \subseteq \R$, we have by Subset implies Cardinal Inequality:

- $\card {\R^+} \le \card \R$

Thus:

- $\card {\powerset \N} \le \mathfrak c$

Hence the result:

- $\mathfrak c = \card {\powerset \N}$

$\blacksquare$

## Sources

- Mizar article TOPGEN_3:29