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Re: [PrimeNumbers] Re: Is 2 a prime

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  • Dr. Michael Paridon
    ... Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT) Von: Phil Carmody An: primenumbers@yahoogroups.com Betreff: Re: [PrimeNumbers] Re: Is
    Message 1 of 15 , Aug 7, 2006
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      -------- Original-Nachricht --------
      Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT)
      Von: Phil Carmody <thefatphil@...>
      An: primenumbers@yahoogroups.com
      Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

      > --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
      > > Sorry, but I do not agree.
      >
      > With what?
      >
      > Please don't top-post, m'kay?
      >
      > Fixing:
      >
      > > Von: Jack Brennen <jb@...>
      > > > Dr. Michael Paridon wrote:
      > > > > I actually think division by zero is not defined.
      > > >
      > > > I would suggest that one can define divisibility by zero without
      > > > needing a definition for division by zero. Say that X is divisible by
      > A
      > > > if there exists any element B such that X = AB.
      > > >
      > > >
      > > > Thus you do not need to define exactly which B represents X/A, only
      > > > that some such B exists. By this definition, zero is divisible by
      > zero.
      > > > Non-zero is not divisible by zero.
      >
      > > Due to correction:
      > >
      > > a) I think divisibility is defined for natural numbers only.
      >
      > But Jack provides a definition which works for N \/ { 0 }.
      > Yes, there exist a handful of simple and convenient definitions which only
      > work
      > for natural numbers, but Jack's wording was pedantically correct - one can
      > provide a definition which does what Jack says it does.
      >
      > > b) You suggested "Say that X is divisible by A if there exists any
      > element B
      > > such that X = AB."
      > >
      > > I think it is "...there exists one and only one distinct element B such
      > that
      > > X = AB." Which of course leads to non-definition of division by zero in
      > all
      > > cases. As a) does, too.
      >
      > That's one possible definition, yes. If Jack were to rely on his
      > definition of
      > divisibility in a paper, I feel sure that he would include that definition
      > if
      > there was any chance of ambiguity.
      >
      > To be deliberately contrary (shock horror!) I would propose that the
      > simplest
      > definition of divisibility is one which doesn't mention division at all,
      > it
      > simply refers to properties of ideals. a|b := (b) \subset (a).
      >
      > You might enjoy working out the divisibility properties of 0 using this
      > definition.
      >
      > Phil
      >
      > () ASCII ribbon campaign () Hopeless ribbon campaign
      > /\ against HTML mail /\ against gratuitous bloodshed
      >
      > [stolen with permission from Daniel B. Cristofani]


      Sorry, if I did not post right. I was not about to cause any inconvenience.

      I did not say Jack's definition is not possible. I think is not the definition mostly used, as usally division by zero is not defined. As well as 0^0, if I remember right. Jack's definition leads to an agreeable result, I admit.

      Best regards

      Michael Paridon



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    • Peter Kosinar
      ... It depends on the area people work in. For a combinatorian, the equality 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number of
      Message 2 of 15 , Aug 7, 2006
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        > Sorry, if I did not post right. I was not about to cause any
        > inconvenience.
        >
        > I did not say Jack's definition is not possible. I think is not the
        > definition mostly used, as usally division by zero is not defined. As
        > well as 0^0, if I remember right. Jack's definition leads to an
        > agreeable result, I admit.

        It depends on the area people work in. For a combinatorian, the equality
        0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
        of functions from empty set to empty set [*]. Moreover, things like
        binomial theorem also work nice with this extension; it allows you to
        evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
        simply 0^n (ok, I admit, this is just a contrived academic example).

        Likewise, if you work in the area of foundations of mathematics, defining
        divisibility using the operation of division is a bit more complicated
        than using the straight existential-quantifer with multiplication (just
        like Jack did); for the division is only a derived operation in e.g. Peano
        Arithmetics and one needs to prove its well-definedness (and possibly some
        other properties) first.

        On the other hand, an analyst would probably bop you over the head
        if he saw 0^0 :-)

        Peter

        [*] This works even in the much more general framework of set-theory --
        If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
        to be the cardinality of the set A^B which is the set of all functions
        from B to A. If the sets A and B are finite, the cardinal exponentation
        agrees with the usual exponentation of natural numbers.

        --
        [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
      • Dr. Michael Paridon
        So I thank everybody very much for explanation. Mathematics has more hidden beauties I will ever learn, and as I am not in a professional way dealing with
        Message 3 of 15 , Aug 7, 2006
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          So I thank everybody very much for explanation. Mathematics has more hidden
          beauties I will ever learn, and as I am not in a professional way dealing
          with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

          Best regrads

          Michael


          ----- Original Message -----
          From: "Peter Kosinar" <goober@...>
          To: "Dr. Michael Paridon" <dr.m.paridon@...>
          Cc: <primenumbers@yahoogroups.com>
          Sent: Monday, August 07, 2006 4:19 PM
          Subject: Re: [PrimeNumbers] Re: Is 2 a prime


          > > Sorry, if I did not post right. I was not about to cause any
          > > inconvenience.
          > >
          > > I did not say Jack's definition is not possible. I think is not the
          > > definition mostly used, as usally division by zero is not defined. As
          > > well as 0^0, if I remember right. Jack's definition leads to an
          > > agreeable result, I admit.
          >
          > It depends on the area people work in. For a combinatorian, the equality
          > 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
          > of functions from empty set to empty set [*]. Moreover, things like
          > binomial theorem also work nice with this extension; it allows you to
          > evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
          > simply 0^n (ok, I admit, this is just a contrived academic example).
          >
          > Likewise, if you work in the area of foundations of mathematics, defining
          > divisibility using the operation of division is a bit more complicated
          > than using the straight existential-quantifer with multiplication (just
          > like Jack did); for the division is only a derived operation in e.g. Peano
          > Arithmetics and one needs to prove its well-definedness (and possibly some
          > other properties) first.
          >
          > On the other hand, an analyst would probably bop you over the head
          > if he saw 0^0 :-)
          >
          > Peter
          >
          > [*] This works even in the much more general framework of set-theory --
          > If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
          > to be the cardinality of the set A^B which is the set of all functions
          > from B to A. If the sets A and B are finite, the cardinal exponentation
          > agrees with the usual exponentation of natural numbers.
          >
          > --
          > [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
          >
          >
          > __________ NOD32 1.1694 (20060805) Information __________
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          >
          >
        • Alan McFarlane
          ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
          Message 4 of 15 , Aug 7, 2006
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            Dr. Michael Paridon wrote:
            > So I thank everybody very much for explanation. Mathematics has more hidden
            > beauties I will ever learn, and as I am not in a professional way dealing
            > with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

            Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction.

            > Best regrads
            >
            > Michael
            >
            >
            > ----- Original Message -----
            [snip]


            --
            Alan
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