Cantor Set is Uncountable

Theorem

The Cantor set $\CC$ is uncountable.


Proof 1

From the definition as a ternary representation, $\CC$ consists of all the elements of $\closedint 0 1$ which can be written without using a $1$.

So let $x \in \CC$. Then in base $3$ notation, we have (as $0 \le x \le 1$):

$\ds x = \sum_{i \mathop = 1}^\infty r_j 3^{-j}$

From the definition of the Cantor set, we have $\forall j: r_j \in \set {0, 2}$.

Furthermore, from Representation of Ternary Expansions, the $r_j$ are unique.


Now define the following function:

$\ds f: \CC \to \closedint 0 1, \quad \map f {\sum_{i \mathop = 1}^\infty r_j 3^{-j} } = \sum_{i \mathop = 1}^\infty \frac {r_j} 2 2^{-j}$

Observe that $\forall j: \dfrac {r_j} 2 \in \set {0, 1}$.

That the right hand side expression is in fact an element of $\closedint 0 1$ now follows from binary notation.


Furthermore by Existence of Base-N Representation, any element $y$ of $\closedint 0 1$ may be written in the following form (where $\forall j: b_j \in \set {0, 1}$):

$\ds y = \sum_{i \mathop = 1}^\infty b_j 2^{-j}$

Obviously, $y = \map f x$, where $x \in \CC$ is defined as follows:

$\ds x = \sum_{i \mathop = 1}^\infty 2 b_j 3^{-j}$


It follows that $f$ is surjective.

From Closed Interval in Reals is Uncountable, the closed interval $\closedint 0 1$ is uncountable.


The result follows.


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$\blacksquare$


Proof 2

It follows from Representation of Ternary Expansions that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 2}$ is an element of $\CC$.

We also have that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 1}$ is an element of $\closedint 0 1 \subset \R$ expressed in binary notation.

Let $f: \CC \to \closedint 0 1$ be the function defined by:

$\forall x \in \CC: \map f x = \text{ the number obtained by replacing every } 2 \text { in } x \text { with a } 1$

where $x$ is expressed in base $3$ notation.

It is clear from the above that $f$ is a surjection.

$\blacksquare$


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cantor set
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