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Re: [PrimeNumbers] Re: Is 2 a prime

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  • Dr. Michael Paridon
    Sorry, but I do not agree. Due to correction: a) I think divisibility is defined for natural numbers only. b) You suggested Say that X is divisible by A if
    Message 1 of 15 , Aug 7, 2006
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      Sorry, but I do not agree.

      Due to correction:

      a) I think divisibility is defined for natural numbers only.

      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.

      Best regards

      Michael Paridon

      -------- Original-Nachricht --------
      Datum: Sat, 05 Aug 2006 09:05:57 -0700
      Von: Jack Brennen <jb@...>
      An: primenumbers@yahoogroups.com
      Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

      > 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.
      >

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    • Phil Carmody
      ... With what? Please don t top-post, m kay? ... But Jack provides a definition which works for N / { 0 }. Yes, there exist a handful of simple and convenient
      Message 2 of 15 , Aug 7, 2006
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        --- "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

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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 3 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 4 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 5 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 __________
              >
              > Diese E-Mail wurde vom NOD32 antivirus system geprüft
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              >
              >
            • Alan McFarlane
              ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
              Message 6 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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