In first-order logic, there exist non-standard countable models of the naturals which are not well-ordered, i.e. there exist sets of naturals with no least element. Hence, not only does Gene's argument fail in first-order logic, but so does the conclusion. Sincerely, Adam P. Goucher http://cp4space.wordpress.com
----- Original Message ----- From: Charles Greathouse Sent: 01/06/14 03:22 PM To: Eugene Salamin, math-fun Subject: Re: [math-fun] Numbers Aplenty
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.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Jan 5, 2014 at 5:10 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
Show us how Gene's argument fails.
-- Gene
________________________________ From: Charles Greathouse <charles.greathouse@case.edu> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun < math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 12:28 PM Subject: Re: [math-fun] Numbers Aplenty
The website looks great!
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
(Easy) exercise: Show how Gene's argument fails in first-order logic. (Hint: Russell.)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Jan 5, 2014 at 3:21 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
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