By G.G. Berry:
Regarding "the smallest integer that cannot be expressed in less than 13 words". The phrase is 12 words long.
Max Black's variation on this paradox goes something like: Consider the the smallest integer that is not referred to in any way on Everything. Since I just referred to it, there can be no such integer.

A Blather of Paradoxes
I think I can explain this one better, or more formally maybe.

Consider 'the least integer not nameable in fewer than nineteen syllables'. This expression appears to name a number (one hundred and eleven thousand, seven hundred and seventy seven, 111777), but it is also an expression of eighteen syllables that is itself a name of a number. So the least integer not nameable in fewer than 19 syllables is nameable by an expression containing fewer than nineteen syllables, which is a contradiction.

This is my favourite paradox, stated by G.G. Berry in 1906.

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