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Shortest Long Number (Posted on 2004-04-15) Difficulty: 3 of 5
What is the smallest positive integer that cannot be defined in less than twenty-five syllables?

  Submitted by Sam    
Rating: 3.0833 (12 votes)
Solution: (Hide)
There are an infinite number of positive integers. There are only a finite number of words in the English language at any one time, and so there are only a finite number of phrases of twenty-four syllables or less. Therefore, the majority of integers must not be specifiable in less than twenty-five syllables.

If we have a set of positive integers, there must be a least member. Therefore, there must be a smallest positive integer not specifiable in less than twenty-five syllables.

But doesn't the specification "the smallest positive integer not specifiable in less than twenty-five syllables" have less than twenty-five syllables?

Therefore, the smallest positive integer not specifiable in less than twenty-five syllables can in fact be specified in less than twenty-five syllables. Thus, the least member of the set isn't really a member of the set after all. Therefore there is no smallest member. Without a smallest member, the entire set collapses.

So does that mean that there are no numbers not specifiable in less than twenty-five syllables? Of course not, or we could only have a finite number of integers. So what's wrong? And is that Gödel I see lurking round the corner again...?

This paradox is known as Berry's Paradox, and was published by Russell in the begining of the last century.

[It has been argued in the comments that the phrase "The smallest positive integer not specifiable..." is not a definition. While I think that this is an interesting angle, I'd argue that "The smallest integer not specifiable in less than two syllables" quite unambiguously defines seven, and so we should expect that this idea could be inducted to twenty-five syllables.]

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: smartalecking the solution (Spoiler)Dej Mar2006-07-08 04:22:02
hhhhhhhmmmmmmmma2006-07-07 06:06:26
Some Thoughtssmartalecking the solutionAdam2005-12-21 23:57:35
re: Possible solution...Will2005-06-09 23:36:55
Possible solution...Will2005-06-09 23:35:44
Godel and GarciadiegoJohn Ryskamp2005-04-24 04:03:52
SolutionFull solutionAngela2005-02-15 12:22:49
No SubjectSam2005-02-03 09:48:30
Excluding the definition argumentJohn2004-07-22 02:22:11
first solution i thuoght ofluke2004-04-29 23:36:23
re: Paradoxes vrs. DilemmasThalamus2004-04-16 10:12:36
Paradoxes vrs. DilemmasPenny2004-04-16 08:12:03
Some ThoughtsBerry's paradox isn't Berry'sFederico Kereki2004-04-16 07:53:22
SolutionWaiting For GodelPenny2004-04-16 06:09:08
re(7): Tentative solution REVISED (reply to Sam)Thalamus2004-04-16 03:40:57
re(6): Tentative solution REVISED (reply to Sam)Sam2004-04-16 02:35:57
Hints/Tipsre: Didn't define it either, AdyAdy TZIDON2004-04-16 02:21:45
Didn't define it either, AdyPenny2004-04-16 00:15:59
Some Thoughtsre: I think I have the solution..-there is none.....Ady TZIDON2004-04-16 00:04:04
re(5): Tentative solution REVISED (reply to Sam)Ady TZIDON2004-04-15 23:50:01
re(3): Tentative solution REVISED / att:samAdy TZIDON2004-04-15 23:43:56
SolutionI think I have the solution.......Penny2004-04-15 23:39:12
Solutionre: maybeJessica2004-04-15 23:07:58
SolutionmaybeJessica2004-04-15 22:59:32
A (good?) referenceRichard2004-04-15 22:40:27
re(4): But can any set of positive integers exist without a least member?Tristan2004-04-15 21:21:44
re(3): But can any set of positive integers exist without a least member?Penny2004-04-15 21:14:26
re(2): But can any set of positive integers exist without a least member?Penny2004-04-15 21:09:53
re: But can any set of positive integers exist without a least member?Sam2004-04-15 20:53:31
But can any set of positive integers exist without a least member?Penny2004-04-15 20:27:28
re(6): Tentative solution REVISED (Paradoxes)Sam2004-04-15 19:17:58
re(5): Tentative solution REVISED (Paradoxes)Penny2004-04-15 18:43:59
re(5): Tentative solution REVISED (Paradoxes)Thalamus2004-04-15 18:41:45
re(4): Tentative solution REVISED (Paradoxes)Sam2004-04-15 18:21:25
re(3): Tentative solution REVISED (Paradoxes)Penny2004-04-15 18:08:16
re(2): Tentative solution REVISED (Paradoxes)Sam2004-04-15 17:50:13
re(5): Tentative solution - DefinitionsRichard2004-04-15 17:39:59
re(4): Tentative solution - DefinitionsThalamus2004-04-15 17:18:02
Some Thoughtsre: Tentative solution REVISEDAdy TZIDON2004-04-15 17:17:59
re(3): Tentative solution - DefinitionsSam2004-04-15 17:11:03
o_0CMB2004-04-15 17:00:44
re(2): Tentative solutionOren Melzer2004-04-15 16:58:45
re: Tentative solutionstan2004-04-15 15:15:22
Bye Bye silly threadPenny2004-04-15 15:09:57
Berry tricky !!!!!!!Penny2004-04-15 14:56:27
SolutionBerry ParadoxOskar2004-04-15 14:47:27
re(2): Tentative solutionPenny2004-04-15 14:46:57
Hints/Tipsre: Tentative solutionsassy2004-04-15 14:45:52
SolutionTentative solutionPenny2004-04-15 14:35:43
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