Powerset is not Subset of its Set/Proof 1
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Theorem
Let $A$ be a set.
Then:
- $\powerset A \not \subseteq A$
Proof
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 $C \in C$.
$\blacksquare$
Sources
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous): Chapter $1$: Exercise $1.2$