3. Russell's paradox

Russell’s paradox

Not every property is suitable for describing sets.

Let $T$ be the set of all sets that are not elements of themselves:

$$T=\{A|A\text{ is a set and }A\notin A\}$$

We need to find out if $T$ is an element of $T$ or not.

• If $T\in T$, then the property used to describe $T$ must hold for $T$ but then $T\notin T$ which is impossible.
• If $T\notin T$, then the property does not hold for $T$, so it is not the case that ”$T$ is a set and $T\notin T$”, which means $T$ is either not a set or $T\in T$. But as $T$ is a set, $T\in T$ which we already proved to be impossible.

title: Not in exam!