Russells's Paradox

Frege's attempt to provide a rigorous set-theoretic foundation for mathematics included his Axiom V, from which Betrand Russell's famously deduced a contradiction as follows.

Axiom V, or the ``axiom of abstraction," is

$$\exists y \forall x (x \in y \leftrightarrow \phi(x)),$$

where the only free variable in sub-formula $\phi$ is x. Substitute `$x \not\in x$' for `$\phi(x)$' to yield

$$\exists y \forall x (x \in y \leftrightarrow x \not\in x).$$

Next, eliminate the existential quantifier and substitute (the not previously used) constant a for y; this produces

$$\forall x (x \in a \leftrightarrow x \not\in x).$$

Finally, eliminate the universal quantifier and substitute the same constant for x to give the contradiction

$$a \in a \leftrightarrow a \not\in a.$$