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semantic distinction

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  • Roger Bagula
    A semantic distinction needs to be made: Infinitely many primes is distinct from an Infinite Prime. Although one seems to imply the other, they really involve
    Message 1 of 16 , Nov 4, 2003
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      A semantic distinction needs to be made:

      Infinitely many primes is distinct from an Infinite Prime.

      Although one seems to imply the other,
      they really involve two separate cases.

      --
      Respectfully, Roger L. Bagula
      tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
      URL : http://home.earthlink.net/~tftn
      URL : http://victorian.fortunecity.com/carmelita/435/
    • Andrew Swallow
      ... BUT WHAT DO YOU MEAN BY AN INFINITE PRIME ? How many times do various people have to ask...? Please be kind enough to explain what you mean by an infinite
      Message 2 of 16 , Nov 4, 2003
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        --- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
        > A semantic distinction needs to be made:
        >
        > Infinitely many primes is distinct from an Infinite Prime.
        >
        > Although one seems to imply the other,
        > they really involve two separate cases.

        BUT WHAT DO YOU MEAN BY AN "INFINITE PRIME"? How many times do various
        people have to ask...?

        Please be kind enough to explain what you mean by an infinite prime.
        To the vast majority of mathematicians, it is absolutely meaningless.
        Primes are by definition finite. If you're going to introduce infinite
        primes, then you must define them properly, not just vaguely say that
        they are something different.

        Andy
      • Nathan Russell
        ... Specifically, one exists, the other does not. The determination of separate cases is a matter of basic boolean algebra - T F = F ... Nathan
        Message 3 of 16 , Nov 4, 2003
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          --On Tuesday, November 04, 2003 08:36:12 AM -0800 Roger Bagula <tftn@...> wrote:

          > A semantic distinction needs to be made:
          >
          > Infinitely many primes is distinct from an Infinite Prime.
          >
          > Although one seems to imply the other,
          > they really involve two separate cases.

          Specifically, one exists, the other does not.
          The determination of separate cases is a matter
          of basic boolean algebra -

          T<->F = F

          :)

          Nathan
        • Paul Leyland
          ... It does indeed. We are making progress. That is precisely the distinction that I ve been trying so hard to make explicit. It is my position and, I
          Message 4 of 16 , Nov 4, 2003
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            > A semantic distinction needs to be made:
            >
            > Infinitely many primes is distinct from an Infinite Prime.
            >
            > Although one seems to imply the other,
            > they really involve two separate cases.

            It does indeed. We are making progress. That is precisely
            the distinction that I've been trying so hard to make
            explicit.

            It is my position and, I believe, that of most mathematicians
            that the term "Infinite Prime" is meaningless.

            I am quite prepared to be convinced that I'm wrong (I've been
            wrong enough times in the past) but to convince me you have to
            explain very clearly what you mean by the term "Infinite Prime"
            --- so clearly that even I can understand you.


            Paul
          • Roger Bagula
            If you have a set of primes that is infinite in number, then how can the last asymptotic element at n*log(n) not be infinite, if n is infinite? And since it is
            Message 5 of 16 , Nov 4, 2003
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              If you have a set of primes that is infinite in number,
              then how can the last asymptotic element at
              n*log(n)
              not be infinite, if n is infinite?
              And since it is an element of the set of primes,
              it is an infinite prime.
              So the proofs which are at least four in number
              that say the primes are infinite sets are either wrong
              or faulted if there is no infinite last element.
              It is a logical problem:
              a paradox.
              My answer has been to set Infinite primes as a metamathematical concept:
              a definition or axiom. You either let the axiom exist or
              not by "choice". You don't get all emotionally
              involved as if it threatened your existence or something.
              That is a "theological" reaction
              not an scientifically detached one.
              Mathematics isn't about "beliefs", but about facts and
              deductions. Mathematics is
              a "tool" for higher understanding.
              One of the better ones, but still not
              the end answer to all knowledge and understanding.
              Philosophy is what tries to do that.

              You seem to be one of the better minds in this group:
              able to stretch your understanding beyond what
              you where taught in school.

              Paul Leyland wrote:

              >> A semantic distinction needs to be made:
              >>
              >>Infinitely many primes is distinct from an Infinite Prime.
              >>
              >>Although one seems to imply the other,
              >>they really involve two separate cases.
              >>
              >>
              >
              >It does indeed. We are making progress. That is precisely
              >the distinction that I've been trying so hard to make
              >explicit.
              >
              >It is my position and, I believe, that of most mathematicians
              >that the term "Infinite Prime" is meaningless.
              >
              >I am quite prepared to be convinced that I'm wrong (I've been
              >wrong enough times in the past) but to convince me you have to
              >explain very clearly what you mean by the term "Infinite Prime"
              >--- so clearly that even I can understand you.
              >
              >
              >Paul
              >
              >
              >

              --
              Respectfully, Roger L. Bagula
              tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
              URL : http://home.earthlink.net/~tftn
              URL : http://victorian.fortunecity.com/carmelita/435/





              [Non-text portions of this message have been removed]
            • pakaran42
              ... There is no last prime. Each prime has a successor, in the same way each integer has a successor. Imagine all the primes as signs along a path. You
              Message 6 of 16 , Nov 4, 2003
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                --- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
                > If you have a set of primes that is infinite in number,
                > then how can the last asymptotic element at
                > n*log(n)
                > not be infinite, if n is infinite?
                > And since it is an element of the set of primes,
                > it is an infinite prime.

                There is no last prime. Each prime has a successor, in the same way
                each integer has a successor. Imagine all the primes as signs along a
                path. You start at 2, walk past 3, and so forth - you can keep
                walking as long as you like, but you will never find one "infinite
                prime," any more than you can walk infinitely far along that path.
                You can just walk any arbitrary distance, or twice that distance, or
                2^1000 times that distance - but not "reach infinity" - there's no
                integer named "infinity" that has no successor.

                Nathan
              • Jud McCranie
                ... One doesn t seem to imply the other (there s no such thing as in infinite prime). We re trying to make sure that you know that.
                Message 7 of 16 , Nov 4, 2003
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                  At 11:36 AM 11/4/2003, Roger Bagula wrote:
                  >A semantic distinction needs to be made:
                  >
                  >Infinitely many primes is distinct from an Infinite Prime.
                  >
                  >Although one seems to imply the other,
                  >they really involve two separate cases.

                  One doesn't seem to imply the other (there's no such thing as in infinite
                  prime). We're trying to make sure that you know that.
                • Jud McCranie
                  ... This gets back to another misconception that I thought you had (I wrote about it last night) (infinite vs. unbounded). What does last asymptotic element
                  Message 8 of 16 , Nov 4, 2003
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                    At 12:34 PM 11/4/2003, Roger Bagula wrote:
                    >If you have a set of primes that is infinite in number,
                    >then how can the last asymptotic element at
                    >n*log(n)
                    >not be infinite, if n is infinite?

                    This gets back to another misconception that I thought you had (I wrote
                    about it last night) (infinite vs. unbounded). What does "last asymptotic
                    element" mean? There is no last prime - Euclid proved that. But just
                    because the size of prime numbers is not bounded that doesn't mean that the
                    size is infinite.

                    It seems that you want to take n=infinity in n*log(n) and you can't do
                    that. The primes are indexed by the positive integers 1, 2, 3,
                    ... There is no prime whose index is infinity. Infinity is not in the set
                    of integers.
                  • pakaran42
                    ... asymptotic ... that the ... the set ... I think part of the confusion here is that we re used to the cardinality of a set of consecutive integers,
                    Message 9 of 16 , Nov 4, 2003
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                      --- In primenumbers@yahoogroups.com, Jud McCranie <j.mccranie@a...> wrote:
                      > At 12:34 PM 11/4/2003, Roger Bagula wrote:
                      > >If you have a set of primes that is infinite in number,
                      > >then how can the last asymptotic element at
                      > >n*log(n)
                      > >not be infinite, if n is infinite?
                      >
                      > This gets back to another misconception that I thought you had (I wrote
                      > about it last night) (infinite vs. unbounded). What does "last
                      asymptotic
                      > element" mean? There is no last prime - Euclid proved that. But just
                      > because the size of prime numbers is not bounded that doesn't mean
                      that the
                      > size is infinite.
                      >
                      > It seems that you want to take n=infinity in n*log(n) and you can't do
                      > that. The primes are indexed by the positive integers 1, 2, 3,
                      > ... There is no prime whose index is infinity. Infinity is not in
                      the set
                      > of integers.

                      I think part of the confusion here is that we're used to the
                      cardinality of a set of consecutive integers, beginning at 1, being
                      the highest integer in that set. This works fine for counting coins,
                      for example. However, it only makes sense for finite sets.

                      A road with 10 mile markers ends at mile marker 10. A road with 500
                      markers ends at mile marker 500. A road that is infinitely long, and
                      has infinitely many mile markers, does not end, and needs no mile
                      marker numbered "infinity". And this is drifting off topic.

                      Nathan
                    • Andrew Swallow
                      ... A logical problem brought about only by the way you choose to present it. If you have a set of primes which is infinite in number, then all it means is
                      Message 10 of 16 , Nov 4, 2003
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                        --- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
                        > If you have a set of primes that is infinite in number,
                        > then how can the last asymptotic element at
                        > n*log(n)
                        > not be infinite, if n is infinite?
                        > And since it is an element of the set of primes,
                        > it is an infinite prime.
                        > So the proofs which are at least four in number
                        > that say the primes are infinite sets are either wrong
                        > or faulted if there is no infinite last element.
                        > It is a logical problem:

                        A logical problem brought about only by the way you choose to present
                        it. If you have a set of primes which is infinite in number, then all
                        it means is that for any number N, there exists an element p of the
                        set such that p>=N. (In fact there exist infinitely many such p, but
                        that doesn't matter too much.) It doesn't have to mean that there
                        exists a "last" element which is infinite.

                        Andy
                      • Jud McCranie
                        ... For one thing, the asymptotic limit of a set does not have to be a member of the set! You are assuming that it is, but that isn t true. In particular,
                        Message 11 of 16 , Nov 4, 2003
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                          >At 12:34 PM 11/4/2003, Roger Bagula wrote:
                          > >If you have a set of primes that is infinite in number,
                          > >then how can the last asymptotic element at
                          > >n*log(n)
                          > >not be infinite, if n is infinite?

                          For one thing, the asymptotic limit of a set does not have to be a member
                          of the set! You are assuming that it is, but that isn't true. In
                          particular, the asymptotic limit of the set of primes isn't even an
                          integer, much less prime.
                        • Jud McCranie
                          ... I don t think it is off-topic, it clearly illustrates what I had been saying.
                          Message 12 of 16 , Nov 4, 2003
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                            At 01:33 PM 11/4/2003, pakaran42 wrote:

                            >A road with 10 mile markers ends at mile marker 10. A road with 500
                            >markers ends at mile marker 500. A road that is infinitely long, and
                            >has infinitely many mile markers, does not end, and needs no mile
                            >marker numbered "infinity". And this is drifting off topic.

                            I don't think it is off-topic, it clearly illustrates what I had been saying.
                          • Joseph Moore
                            1) Infinitely many primes. This is easy (read: euclidean proof...). 2) An infinite prime. I ll make up a definition, just for the fun of it: Let
                            Message 13 of 16 , Nov 4, 2003
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                              1) Infinitely many primes. This is easy (read:
                              euclidean proof...).

                              2) An 'infinite' prime. I'll make up a definition,
                              just for the fun of it:

                              "Let p(x,n)=sum(i=0, n, ai*x^i)=a0+a1*x+a2*x^2+... be
                              a polynomial in the polynomial ring over the integers
                              such that for all n, p(x,n) is irreducible in this
                              ring. Then we say that p(x)=sum(i=0, infinity,
                              ai*x^i) is irreducible."

                              So this could be a potential definition of an
                              'infinite prime', although it doesn't seem to have any
                              immediate use...

                              Joseph.


                              --- Roger Bagula <tftn@...> wrote:
                              > A semantic distinction needs to be made:
                              >
                              > Infinitely many primes is distinct from an Infinite
                              > Prime.
                              >
                              > Although one seems to imply the other,
                              > they really involve two separate cases.

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                            • ctcard_hotmail_com
                              ... Well I wouldn t go that far. Here s a definition: If K is a field, then a prime of K is an equivalence class of valuations of K. An infinite prime is a
                              Message 14 of 16 , Nov 4, 2003
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                                "Andrew Swallow" <umistphd2003@...> wrote:
                                >Please be kind enough to explain what you mean by an infinite prime.
                                >To the vast majority of mathematicians, it is absolutely meaningless.
                                Well I wouldn't go that far.
                                Here's a definition:
                                If K is a field, then a prime of K is an equivalence class of
                                valuations of K. An infinite prime is a prime of K that consists of
                                archimedean valuations.
                                (see http://planetmath.org/encyclopedia/InfinitePrime.html for more
                                details, for example)
                                Now, I'm not claiming that it's anything to do with what Roger is on
                                about, but it is a well-known definition of "infinite prime".
                                :-)
                                Chris
                                [ apologies if this appears twice - I tried to post this over 12 hours
                                ago, but it didn't appear ]
                              • Andy Swallow
                                ... If I was feeling immensely cheeky, I d point out that they re also known as infinite places , the substition of primes for places being called by some
                                Message 15 of 16 , Nov 5, 2003
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                                  > Here's a definition:
                                  > If K is a field, then a prime of K is an equivalence class of
                                  > valuations of K. An infinite prime is a prime of K that consists of
                                  > archimedean valuations.
                                  > (see http://planetmath.org/encyclopedia/InfinitePrime.html for more
                                  > details, for example)

                                  If I was feeling immensely cheeky, I'd point out that they're also known
                                  as "infinite places", the substition of "primes" for "places" being called
                                  by some an abuse of notation (from Swinnerton-Dyer's Brief guide to
                                  algebraic number theory, p33)

                                  But sorry, that's just nit picking. Perfectly true, and probably shows
                                  up the gaps in my knowledge, the concept of infinite prime does have a
                                  place in some parts of mathematics. But *not* as the limiting point of
                                  the rational primes, which is what is being suggested in this thread.

                                  But it's all good fun, :-)

                                  Andy
                                • Paul Leyland
                                  From: Roger Bagula [mailto:tftn@earthlink.net] ... Ah, now that one I can answer. You make the assumption that there is a last element. You have not proved
                                  Message 16 of 16 , Nov 5, 2003
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                                    From: Roger Bagula [mailto:tftn@...]

                                    > If you have a set of primes that is infinite in number, then how can
                                    > the last asymptotic element at n*log(n) not be infinite, if n is
                                    > infinite?

                                    Ah, now that one I can answer. You make the assumption that there is
                                    a last element. You have not proved this statement, you just assume
                                    it. I assert that there is no such last element. The proof goes as
                                    follows:

                                    You now seem to accept that the set of primes is infinite, that is it
                                    has cardinality aleph_0. In that case, each and every element of the
                                    set can be labelled uniquely with an integer; this is the definition
                                    of a set which has cardinality aleph_0. So, for any given N, no
                                    matter how large, we can find the prime labelled by N. But, given
                                    such a N, we can find another integer M which is larger than N. An
                                    example of such an M is N+1. Therefore given a prime, no matter how
                                    large, we can find its label AND we can find a larger prime which has
                                    a larger label. Consequently, the sequence of primes is never ending;
                                    we can always find larger and larger primes and we never reach the
                                    last one. The last prime does not exist.

                                    > And since it is an element of the set of primes, it is an infinite
                                    > prime. So the proofs which are at least four in number that say the

                                    And as it doesn't exist in the first place, all your subsequent
                                    reasoning is meaningless.


                                    Paul
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