Russell's Antimony

set-theory

Definition

Russell's Antimony

Let $S$ be the set of all sets. Every element of the power set $\mathcal{P}(S)$ is, by definition, a subset of $S$. Since every subset of $S$ is itself a set, it must, by definition of $S$, be an element of $S$. Therefore, we can conclude that

$$\mathcal{P}(S)\subseteq S$$

Based on that, we have found a surjective function $f:S\to \mathcal{P}(S)$ defined as follows:

• $x\mapsto x$ for all $x$ that are subsets of $S$
• To those elements of $S$ that are not subsets of $S$, we assign the empty set $\varnothing$, which is guaranteed to be an element of $\mathcal{P}(S)$.

This appears to be a contradiction as Cantor’s Theorem states no set can be mapped surjectively onto its power set.

$$A=\{x\in S:x\notin f(x)\}=\{x:x\notin x\}\cup \{x:x⊈S\}$$

Examine the first set $R$ further:

$$R=\{x\in S:x\notin x\}$$

This constitutes Russell’s Paradox. The collection of all sets that do not contain themselves as an element lead to a direct contradiction when we ask: Does this set $R$ contain itself, or not?.

• By definition, an object $x$ is an element of $R$ iff $x$ satisfies $x\notin x$.

• Since this condition must hold for every $x$ in the domain $S$, it must specifically hold for the case where $x=R$:

R \in R \iff R \not\in R