Russell's Paradox
Why is this set theory “Naive”? The reason is that is suffers from paradoxes.
A barber is the man who shaves all those, and only those, men who do not shave themselves.
- Who shaves the barber?
This statement and question form a paradox.
As yet we know that you should always write a predicate like so:
\[\{x\in A \vert x \text{ satisfies some property}\}\]Before Russell sets were just defined by their properties.
Russell’s Paradox shows that the ‘object’ $\{x\vert P(x)\}$ is not always meaningful.
Set $A=\{B\vert B\notin B\}$
Problem: do we have $A\in A$?
Abbreviate, for any set $C$, by $P(C)$ the statement $C\notin$. Then $A=\{B\vert P(B)\}$.
- If $A\in A$, then (from the definition of $P$), not $P(A)$. Therefore $A\notin A$.
- If $A\notin A$, then (from the definition of $P$), $P(A)$. Therefore $A\in A$.
This is the same contradiction that was reached in the opening statement.
This is the reason why we must draw the elements from another set to avoid this paradox. For example, in the definition of even numbers the predicate draws from the set of natural numbers:
\[x\in \mathbb{N} \vert x=2k\ \text{for some int. }k\]