Berry’s Paradox
G. G. Berry, via Bertrand Russell · 1908
The puzzle
“The smallest positive integer not definable in fewer than twelve words.”
Note
The phrase has eleven words, and it appears to define such a number. The paradox is named for G. G. Berry, a junior librarian at the Bodleian who proposed it to Russell in correspondence. It’s a linguistic cousin to Russell’s set-theoretic paradox: both depend on self-reference and unrestricted comprehension. The repair — that “definable” must be relativised to a fixed formal language — works for mathematics but is awkward in natural English, where what counts as defining something is itself vague. Berry was a bookish autodidact who never published any mathematics in his lifetime, but the paradox he sent to Russell still appears in introductory logic textbooks more than a century later. A library clerk’s footnote outlived most of the period’s formal philosophy.