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Re: [PrimeNumbers] Re: Is 2 a prime

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  • 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 1 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
      >
      >
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      >
    • Alan McFarlane
      ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
      Message 2 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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