Let N be the smallest positive integer that cannot be defined
in English using fewer than one hundred thirty English symbols.

12345678901234567890123456789012345678901234567890123456789012345
67890123456789012345678901234567890123456789012345678901234567890

Since there are only finitely many strings in English that use fewer than one hundred thirty English symbols, there can be only finitely many positive integers that can be defined in English using fewer than one hundred thirty English symbols. There are infinitely many positive integers altogether, so the set of positive integers that cannot be defined in English using fewer than one hundred thirty English symbols, which we call S, is non-empty. Every non-empty set of positive integers has a smallest element. We are simply defining N to be the smallest element of S.

But wait!

We defined N using fewer than one hundred thirty English symbols. Therefore, N cannot be in S.

Paradox!