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13955Re: [PrimeNumbers] Infinite primes-> a Turing Machine prime sieve that never stops?

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  • Roger Bagula
    Nov 4, 2003
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      The concept of "infinity"
      is a Platonic ideal.
      It can't be proven.
      It is a limiting axiomatic definition (asymptotic).
      So arguing if such an such types of infinity
      exist , makes sense only if the limiting
      case is assumed to exist in the first place.
      Mathematics can be defined as two cases:
      1) numbers where infinity is defined
      2) numbers where no infinity is defined or definable as
      being a non-operational definition
      ( it crashes Mathematica, ha, ha... )
      It is a philosophical distinction outside of normal mathematics.
      Called since Gödel's time "metamathematical" statements/ arguments
      after it was realized that a philosophical threshold
      existed in certain kinds of proofs.
      Statements like :
      3) Aleph1=2^Aleph0
      are outside of ordinary mathematics.
      The algebra of transfinites is really a form of
      metamathematics invented and outside ordinary mathematics.
      It has it's own axiom system and rules.
      The idea of an infinite prime would have to be
      in this context and not in the context of finite primes at all.
      A lot of people don't like it pointed out that irrational numbers depend on
      a statement like 3)
      being accepted as an axiomatic definition.
      Like Euclid's product proof,
      Cantor's proof of transfinites is also widely faulted in
      modern mathematics.
      The result is you get philosophical
      and nearly religious in tone arguments
      in mathematical forms like these egroups.
      At first I didn't realize that it was that basic a
      "postulate" , "axiom"
      "definition", but it has become clear
      that it is.
      In modern science we are talking about a quantum
      measure limit near 10^(-33) cm, below which
      what we know of as material and fields can't exist.
      Much of current discussion on quantum gravity
      comes down to "finite lattices" of spatial points.
      So ideas at the integer level become very important.
      And "infinities" become important on a philosophical
      do they exist
      or don't they?
      Most of our calculations that make up
      our ordinary lives don't require that any kind of infinity exists.

      I don't know if any of this answers you list of questions or not.
      I really don't like to get in such discussions,
      since math people seem to ignore any philosophical issues by defining
      them away.
      Axioms and definitions are an answer to all their thinking problems?
      As a physical scientist ( chemist, physical scientist)
      I'm not bound by those rules in my thinking.
      So I'm glad to say I can think of things outside of those rules.
      I used to think that the topological approach was an answer
      but that approach is bounded by set theory as well
      and the logical problems of set theory seem to crop up
      in it's proofs of infinite primes too?

      The basic/ fundamental question seems to be :
      Is any system that is based on axioms that resolve into
      Platonic ideals, actually inductively defensible
      as a form of knowledge?

      In other words is science more fundamental
      philosophically than mathematics at a metamathematical level?
      Paul Leyland wrote:

      >>From: Roger Bagula [mailto:tftn@...]
      >>The sequences related to Euclid's proof are:
      >>A000945 A000946 A002585 A005265 A005266 A051342
      >>( there are several new ones as well)
      >>A00945 is just the first "related" sequence.
      >>But there have to be two schools of thought
      >>since "proof" isn't working.
      >>1) Euclid's school infinite primes exist
      >>2) modern school of thought : infinite primes don't exist
      >>Unless you can "conclusively" prove no infinite prime exists...
      >>I've seen nothing like that in the posts.
      >>It can be "argued" both ways. Paul Leyland
      >>is good at arguing, but lacks conclusive proofs to go
      >>with it.
      >Very well, time for another argument. It would be helpful if
      >you respond (concisely and politely please) to each of the
      >questions below.
      >First my thesis: infinite primes do not exist. The set of
      >primes is infinite (because it has cardinality aleph_0).
      >Consider carefully the difference between these two statements.
      >The first says that, because infinity is not an integer and primes
      >are integers, the concept "infinite prime" is meaningless. The
      >second says that the set of primes can be put in one-to-one
      >correspondence with the set of integers >0.
      >Yuo do not seem to like the classical proof, so let's prove the
      >number of primes (i.e the cardinality of the set of primes) is
      >infinite by another approach.
      >Do you agree that the set of natural numbers is infinite?
      >That is, do you believe that given a particular integer N, we can
      >always find a larger integer M, no matter how large N may be?
      >Do you agree that, given N, the value M=N+1 is larger than N?
      >Again, if not the game is over because we are not talking about
      >mathematics as understood by almost all the world's mathematicians.
      >Do you agree that each and every integer N > 1 has a unique
      >factorization into primes?
      >I'm concerned only with which primes appear in the factorization,
      >not with the order in which they are given. Numerical order is
      >conventional but it really doesn't matter which order they are in
      >for the arguments below.
      >Next, I am going to define F_N, where N is an integer > 0, to be
      >the value 2^(2^N)+1, so F_1 has the value 5, F_2 has the value 17,
      >F_3 has the value 257, F_4 has the value 65537, and so on.
      >Do you agree that the set of all values F_N is infinite?
      >I claim it is because the N can take any integer value and each
      >value of N yields a value for F_N.
      >Do you agree that all the values for F_N are distinct?
      >I claim they are because 2^N is larger than N, and 2^(2^N) is
      >larger than 2^N. I therefore claim that each F_i is greater
      >than F_j whenever i>j.
      >Do you agree that (x+y)*(x-y) = x^2 - y^2?
      >Now, let's take a look at F_N - 2. By definition of F_N, this
      >quantity is equal to 2^(2^N) - 1. Note that 1 is a square (it is
      >1 squared and that 2^(2^N) is a square (it is 2^(2^(N-1)) squared.)
      >So, do you agree that the following equation is correct?
      >F_N - 2 = (2^(2^(N-1)) + 1) * ( 2^(2^(N-1)) - 1)
      >If so, you will readily conclude that the first term is just the
      >definition of F_(N-1), and the second is just F_(N-1) - 2).
      >Now what does this tell us? (A rhetorical question, no need to
      >answer). It says that F_N - 2 is a multiple of F_(N-1). Of course,
      >we have to be careful. F_i is not defined if i is less than 1, so
      >we have to insist that N >= 2 in the above equation. So lets start
      >with N=2 and work our way up.
      >When N=2, F_2 - 2 = F_1 * (F_1 - 2).
      >This is easy to check:
      >F_2 is 17, F_1 is 5 and, indeed F_2-2 = 15 = 5*3 = F_1 * (F_1 -2).
      >When N=3, F_3 - 2 = 255
      > = F_2 * (F_2 - 2)
      > = 17 * F_1 * (F_1 - 2)
      > = 17 * 5 * 3.
      >When N = 4, F_4 - 2 = 65535
      > = F_3 * (F_3 - 2)
      > = 257 * F_2 * (F_2 - 2)
      > = 257 * 17 * F_1 * (F_1 - 2)
      > = 257 * 17 * 5 * 3.
      >With me so far? Do you agree that we can continue this process
      >as far as we wish, because we can increase the value of N as far as
      >we wish?
      >If not, why not? I can try to make it clearer. If you do agree,
      >let's proceed.
      >I now claim that F_M has no prime factors in common with *any* F_N
      >for which N < M. The reasoning is that F_M - 2 is a multiple of
      >all F_N for which N < M. Thus, any prime factor of F_N yields a
      >remainder of 2 when divided into F_M. All the F_N numbers are odd,
      >so we can discount the prime number 2 as being a factor of any of them.
      >Do you agree with this claim? If not, please explain what you think
      >is wrong with it.
      >Assuming you do agree, you will also agree that I have produced an
      >infinite set of numbers, none of which share a common prime factor.
      >The set of all F_N consists only of integers, so each and every one
      >has at least one prime factor, agreed?
      >But as the set of all F_N has cardinality aleph_0, the same as the
      >cardinality of the set of all integers >=1, AND every member of the
      >set of all F_N is an integer, AND every member of the set of all F_N
      >has at least one prime factor, AND no two members share any prime
      >factors, THEN the cardinality of the set of all prime factors of all
      >the members of the set of all F_N is itself aleph_0.
      >What does the above lengthy and pedantically stated sentence say
      >(another rhetorical question)? It says that we have constructed
      >a set of prime numbers which has cardinality aleph_0. That is,
      >we've found an infinite set of prime numbers. This, I claim, proves
      >my thesis given at the start of this article. Note I do *not* claim
      >that the set of primes I constructed contains all the primes. In fact,
      >we know for certain that it does not, and you should be able to prove
      >that, for instance, neither 2 nor 13 are members of the set.
      >Nonetheless, it is an infinite set.
      >Conclusion: the set of all primes is infinite (has cardinality
      >aleph_0) because at least one set which contains only primes itself
      >has cardinality aleph_0.
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      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/

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