Power Set of Natural Numbers has Cardinality of Continuum/Proof 1

Theorem

Let $\N$ denote the set of natural numbers.

Let $\powerset \N$ denote the power set of $\N$.

Let $\card {\powerset \N}$ denote the cardinality of $\powerset \N$.

Let $\mathfrak c = \card \R$ denote the cardinality of the continuum.

Then:

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

Proof

Outline

$\powerset \N$ is demonstrated to have the same cardinality as the set of real numbers.

This is done by identifying a real number with its basis expansion in binary notation.

Such a basis expansion is a sequence of $0$s and $1$s.

Each $1$ of a real number $x$ expressed in such a way can be identified with the integer which identifies the power of $2$ representing the $1$ in that place.

Hence $x$ corresponds to the subset of $\N$ whose elements are the positions in $x$ which hold a $1$.

Thus $\R$ and $\powerset \N$ are equivalent.

From the Cantor-Dedekind Hypothesis, the real numbers are in one-to-one correspondence with the points on a line segment.

The points on a line segment form a continuum.

This theorem requires a proof. In particular: Formalise the above You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.