Thursday, November 13, 2008

Berry's Paradox

The natural numbers are one, two, three and so forth. They can grouped by the number of syllables it takes to say them: {one, two, three, ...}, {seven, thirteen, fourteen, ...}, {eleven, seventeen, twenty-one, ...}, ... . So we can easily find, for example, the smallest natural number that cannot be given to us, in that way (by saying its English name), in fewer than three syllables; it is eleven.
......Let us now allow any method of verbally specifying a natural number—e.g. “two hundred and thirty-four times fifty thousand and fifty” (saving us six syllables on simply saying its name)—and consider the following specification:
(B)......The smallest natural number that cannot be specified in less than twenty-five syllables.
Berry’s paradox (which is over a hundred years old) is that (B) is itself only twenty-four syllables long. It seems paradoxical because we expect (B) to specify a number, in at least twenty-five syllables (much as earlier we got eleven, in three syllables), and yet whichever number it is has therefore been given to us by (B) in less than twenty-five syllables.
......However, since we have allowed any method of verbally specifying a natural number, we have allowed that I might take any number and say it at a certain time and date and then refer to it (and hence specify it) in less than twenty-five syllables as, for example, “The number mentioned by me between two and two thirty on the first of June two thousand and eight.” It is quite indeterminate which numbers one might do that for, and hence the number specified by (B) is also indeterminate.
......And if we don’t generalise completely, but instead limit to some extent the ways in which the numbers considered by (B) may be specified, then (B) will either give us some specific number with no problem (much as the specification above gave us eleven), or else it will be rather more obviously self-contradictory (much as Richard's paradox is), so that we would hardly expect it to give us a number (whence it should stop appearing paradoxical).

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