The Berry paradox is just the same: for some fixed definition of definitions-from-words you can find the smallest natural number not among them, but the problem then makes use of a higher-order predicate making use of the original definitions-from-words. Charles Greathouse Analyst/Programmer Case Western Reserve University On Mon, Jan 6, 2014 at 11:00 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Charles, I understand your first paragraph, which I think nails the interesting-number (IN) paradox.
But what is your unraveling of the Berry paradox, and why is it equivalent to IN?
--Dan
On 2014-01-06, at 7:22 AM, Charles Greathouse wrote:
It's a sleight of hand, an antinomy. If you fix any definition of Interesting(x) for which not Interesting(n) for some natural numbers n, you can find the least natural number which is not Interesting. But then you move to a higher-order Interesting'(x) which is based on Interesting(x).
In particular this is an instantiation of the Berry paradox ("the smallest positive integer not definable in fewer than twelve words") first published by Russell, and not too dissimilar to Russell's paradox itself.
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