“The ways of paradox”: Quine on Berry’s paradox.

“Ten has a one-syllable name. Seventy-seven has a five-syllable
name. The seventh power of seven hundred seventy-seven has a name
that, if we were to work it out, might run to 100 syllables or so;
but this number can also be specified more briefly in other terms. I
have just specified it in 15 syllables. We can be sure, however, that
there are no end of numbers that resist all specification, by name or
description, under 19 syllables. There is only a finite stock of
syllables altogether, and hence only a finite number of names or
phrases of less than 19 syllables, whereas there are an infinite
number of positive integers. Very well, then ; of those numbers not
specifiable in less than 19 syllables, there must be a least. And
here is our antinomy : the least number not specifiable in less than
nineteen syllables is specifiable in 18 syllables. I have just so
specified it.
The antinomy belongs to the same family as the antinomies that have
gone before. For the key word of this antinomy, “specifiable”, is
interdefinable with “true of”. It is one more of the truth locutions
that would take on subscripts under the Russell-Tarski plan. The
least number not specifiable-0 in less than nineteen syllables is
indeed specifiable-1 in 18 syllables, but it is not specifiable-0 in
less than 19 syllables ; for all I know it is not specifiable-0 in
less than 23.”

By the way, it seems that Russell thought Berry’s number was 111,777.

  1. Terren says:

    111777 = three hundred thirty four squared plus two hundred twenty one (14 syllables)

    also, forty eight cubed plus one thousand one hundred eighty five (also 14)

  2. Terren says:

    Even better:

    two thousand one hundred nine times fifty three (11)

    nine thousand sixty three times thirty seven (11)

  1. There are no trackbacks for this post yet.

Leave a Reply