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Re: [PrimeNumbers] A classic [s]p[r]oof; Answered.

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  • Chris Caldwell
    ... Like Chris, I thought these were rather old, but I spent some time in the library and it turns out that perhaps they are not. The originator seems to be
    Message 1 of 9 , Mar 6, 2001
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      At 07:47 PM 2/27/01 -0500, Chris Nash wrote:
      > >Do any of you know the source of this theorem or a good
      > >reference that discusses the error in the proof?
      >
      >It is indeed classical, and probably goes back a long long way (long
      >before Russell's formulation). These sort of self-referential

      Like Chris, I thought these were rather old, but I spent some time in
      the library and it turns out that perhaps they are not. The "originator"
      seems to be G. G. Berry of the Bodleian library who wrote the
      paradox (in a messier form) to Russel 21 December 1904. So this
      indeed came after the big set of all sets paradox (and the largest
      cardinal paradox, Richard's least undifinable cardnal paradox and
      the rest). Russel was the first to publish it in 1906 (Berry never
      did). In 1908 Russel published it in this simplified form:

      Berry's paradox:
      Hence "the least integer not nameable in fewer
      than nineteen syllables" must denote a definite
      integer; in fact, it denotes 111,777. But "the
      least integer not nameable in fewer than
      nineteen syllables" is itself a name consisting
      of eighteen syllables; hence the least integer
      not nameable in fewer than nineteen syllables
      can be named in eighteen syllables, which is a
      contradiction.

      Berry introduced himself to Russell with a card that on one side
      said "the statement on the other side of this card is false" and
      on the other side said "the statement on the other side of this
      card is true"--another (now) well known paradox that he also
      originated.

      Berry's paradox is not isomorphic to the Curios (or interesting)
      paradox, but looks to me like the forerunner.

      >statements get a lot of treatment in G\"odel, and so the "error in the
      >proof" is that incompleteness allows us to construct the statement in
      >the first place.

      The Berry type of paradox is a "semantic paradox," and unlike
      Russell's set of all sets and the various diagonal arguments
      (Richards, G\"odel, ...) in that they can not be directly
      expressed within logic. And indeed forms of them are old
      (so the Chris' weren't completely off base!) The first is
      perhaps Epimenides the Cretan's all Cretan's are liars...
    • d.broadhurst@open.ac.uk
      ... Oh, I bet long before some hunter-gather s grunted along these lines round a fire: Ugg: Caught summat? Glugg: Two no-names! Can t see that the curios thing
      Message 2 of 9 , Mar 6, 2001
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        Chris Caldwell wrote:

        > The first is perhaps Epimenides

        Oh, I bet long before some hunter-gather's grunted along
        these lines round a fire:

        Ugg: Caught summat?
        Glugg: Two no-names!

        Can't see that the curios thing is much different from
        the self-contradictory "Anon" appended to a pome....

        David
      • Chris Caldwell
        ... No doubt. But my context is simpler, I was looking for references; and the first seems to be Epimenides. But I d be glad to reference the hunter
        Message 3 of 9 , Mar 7, 2001
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          At 01:35 AM 3/7/01 +0000, d.broadhurst@... wrote:
          >Chris Caldwell wrote:
          > > The first is perhaps Epimenides
          >
          >Oh, I bet long before some hunter-gather's grunted along
          >these lines round a fire:

          No doubt. But my context is simpler, I was looking for references;
          and the first seems to be Epimenides. But I'd be glad to reference
          the hunter gatherers if you know of a cave wall I can cite!

          anon.
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