Cantor's Theorem (Strong Version)/Proof 2/Induction Step/Proof 2

Theorem

Let $S$ be a set.

Let $\map {\PP^n} S$ be defined recursively by:

$\map {\PP^n} S = \begin{cases} S & : n = 0 \\ \powerset {\map {\PP^{n - 1} } S} & : n > 0 \end{cases}$

where $\powerset S$ denotes the power set of $S$.

Then $S$ is not equivalent to $\map {\PP^n} S$ for any $n > 0$.

Induction Step

This is our induction hypothesis:

There is no surjection from $S$ onto $\map {\PP^k} S$.

We are to show:

There is no surjection from $S$ onto $\map {\PP^{k + 1} } S$.

Proof

By Power Set is Nonempty, $\powerset {\map {\PP^k} S}$ is non-empty.

By definition:

$\map {\PP^{k + 1} } S = \powerset {\map {\PP^k} S}$

Then:

$\map {\PP^{k + 1} } S \ne \O$

By Law of Excluded Middle, there are two choices:

$S = \O$

or:

$S \ne \O$

Suppose that $S = \O$.

By Image of Empty Set is Empty Set: Corollary 2, there is no surjection from $S$ onto $\map {\PP^{k + 1} } S$.

Suppose that $S \ne \O$.

By the induction hypothesis $\map P k$ is true.

Aiming for a contradiction, suppose that $f: S \to \map {\PP^{k + 1} } S$ is a surjection.

By Injection from Set to Power Set, there is an injection:

$g: \map {\PP^k} S \to \powerset {\map {\PP^k} S}$

By Injection has Surjective Left Inverse Mapping, there exists a surjection:

$h: \powerset {\map {\PP^k} S} \to \map {\PP^k} S$

By definition of $\powerset {\map {\PP^k} S}$, this is a surjection:

$h: \map {\PP^{k + 1} } S \to \map {\PP^k} S$

By Composite of Surjections is Surjection, we have a surjection:

$h \circ f: S \to \map {\PP^k} S$

But this contradicts the induction hypothesis.

Thus we conclude that the theorem holds for all $n$.

$\blacksquare$