Powerset is not Subset of its Set
Theorem
Let $A$ be a set.
Then:
- $\powerset A \not \subseteq A$
Proof 1
Aiming for a contradiction, suppose that $\powerset A \subseteq A$, and define:
- $C = \set {x \in \powerset A : x \notin x}$
We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$.
Since $\powerset A \subseteq A$, we have:
- $C \subseteq A$
and thus
- $C \in \powerset A$
We can derive a similar contradiction to Russell's Paradox.
If $C \in C$, then it must meet $C$'s condition that $C \notin C$.
If $C \notin C$, then it meets $C$'s condition for $C \in C$.
$\blacksquare$
Proof 2
Aiming for a contradiction, suppose that $\powerset A \subseteq A$.
Let $I: \powerset A \to A$ be the identity mapping.
$I$ is an injection by Identity Mapping is Injection.
But by No Injection from Power Set to Set, this is a contradiction.
$\blacksquare$
Proof 3
Aiming for a contradiction, suppose that $\powerset A \subseteq A$.
Since $A \in \powerset A$, this implies:
- $A \in A$
But this contradicts Set is Not Element of Itself.
$\blacksquare$