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Whole prime number

  1. Jul 28, 2007 #26

    Hurkyl

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    From the point of view of formal logic, having a multiplication operation makes a rather large difference.

    If we use first-order logic, and we omit general multiplication, we have the theory of Presburger arithmetic. This theory is known to be logically complete. But if we use Peano arithmetic... or even if we omit the induction axiom and use Robinson arithmetic, then we are working with an incomplete theory.

    In other words, if we stick only to addition, every (first-order) proposition we can state about the natural numbers can either be proven or disproven. But if we allow multiplication, then there exist statements that can neither be proven or disproven. (And furthermore, it remains incomplete, even if we adopt finitely more axiom schema)


    I don't know much about what happens when you allow higher-order logic.
     
    Last edited: Jul 28, 2007
  2. Jul 28, 2007 #27

    Hurkyl

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    Well, of course Peano was thinking about the "number line". His goal (I presume) was to write down a list of axioms that characterized the intuitive notion of "natural number" that he and other mathematicians had.

    The modern approach to mathematics prefers to have explicit foundations -- these days, one often defines that anything satisfying the Peano axioms is a "set of natural numbers", or similar. Then, if we turn to metamathematics, we argue that the counting process does, in fact, satisfy the Peano axioms, and so we are justified in saying that when we count, we are using the natural numbers.
     
  3. Jul 28, 2007 #28

    CRGreathouse

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    Presburger arithmetic may be of interest to philiprdutton because it's just 'on the edge' of being able to define primes. It can't define the concept of being prime in general, let alone prove statements about them, but it can (I think) show that particular numbers are prime:

    [tex]\neg\exists x : x + x = 5[/tex]
    [tex]\neg\exists x : x + x + x = 5[/tex]
    [tex]\neg\exists x : x + x + x + x = 5[/tex]
     
  4. Jul 28, 2007 #29
    mapping


    Finally I had one last important thought. Given the system described where you have the natural numbers but you can not add or take the successor, we should be able to map the system to the system where by you can build the natural numbers. If such a mapping is "formalized" then the problem appears. On the one hand you have a system where "prime" is not defined and on the other hand you have a system where "prime" can be defined. They are mapped to each other and so now is there a paradox?

    I am thinking about the utility of mapping similar to what is used by Godel in his famous proof.

    Again for clarity: we defined a "counting" style, infinite statement axiomatic system which you have no notion of multiplication nor successor function (as in the above posts). We have another system like Peano. Both systems produce something that lies on the same place on the number line. We use mapping to link the two systems through the "number line." Now, despite the mapping (if it is possible), you can not impose the notion of prime on the simpler system. Hence, the notion of "prime" is directly related to the mechanisms of addition/multiplication or other operations.... NOT the actually position on the number line thing.
     
  5. Jul 28, 2007 #30
    "The idea of a prime number is loads of fun for the guy with all the numbers AND the bag of tools with which he can do things to those numbers. The guy with only all the numbers is simply bored out of his mind." - Philip Ronald Dutton

    (sorry! I am exploiting the utility of writing hoping it will sharpen my understanding of all this)
     
  6. Jul 28, 2007 #31

    CRGreathouse

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    But of course. I can also set up a linking from the "counting" (on the left) to Peano Arithmetic (on the right) like so:

    1 <--> 3
    2 <--> 2
    3 <--> 1
    4 <--> 6
    5 <--> 5
    6 <--> 4
    7 <--> 9
    . . .

    The mapping is perfectly reasonable, and all properties (i.e. none) that held in the counting system still hold in Peano arithmetic. The counting numbers that are prime in PA, though, are 1, 2, 5, and so on -- not at all the same.
     
  7. Jul 28, 2007 #32
    branches


    May I ask how you start with 3 and then get 2.... 1,6,5,4,9?

    Also, is the counting system single branch (of statements) as opposed to a multi-branch PA tree of types of statements? If each axiom in PA can produce a certain amount of statements then that set of statements is what I am informally calling a branch. Since the counting system only has one way to make statements it is single branch.
     
  8. Jul 29, 2007 #33
    If you want to play "da da da da" for a while, stress one "da" of every N, as in "da da DA da da DA ..."; if you put them all together,

    2 da DA da DA da DA da DA da DA da DA da ...
    3 da da DA da da DA da da DA da da DA da da DA ...
    4 da da da DA da da da DA da da da DA da da da DA ...
    5 da da da da DA da da da da DA da da da da DA da da da da DA ...
    6 da da da da da DA da da da da da DA da da da da da DA ...
    7 da da da da da da DA da da da da da da DA da da da da da da DA ...


    a prime number is one where the first stressed DA's won't coincide with any DA for all smaller numbers.

    (Which of course is a re-edition of the [/PLAIN] [Broken]
    Sieve of Eratosthenes
    .)
     
    Last edited by a moderator: May 3, 2017
  9. Jul 29, 2007 #34

    CRGreathouse

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    Yes, but that would be beside the point. They have no properties, so there's nothing making the counting "2" more or less like the Peano "2" than the Peano "7". I can put them in any order I want -- and in fact I could associate them with only the Peano primes, or only the Peano composites, or only the Peano powers of 2 that are squares.

    Terminology. Remember that both Peano arithmetic and your counting system have an infinite number of axioms -- you have one axiom schema, which actually has omega members (one for each natural number). So yes, each of your axioms has only one statement it can make, but you can make an infinite number of statements.

    That aside, I'm still not sure I quite follow. What is the motivation behind the branch terminology?
     
  10. Jul 29, 2007 #35

    Hurkyl

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    Incidentally, as far as formal logic is concerned, the axiomatic method is merely a convenient way for presenting formal theories. There is no inherent quality of a statement that determines whether or not it is an axiom -- it's simply an artifact of the way the formal theory is presented.
     
  11. Jul 29, 2007 #36
    creating a number system

    Okay, so basically you just created a numbering system. Given the counting system you just added some machinery to give you the ability to talk about where you stopped counting. Once you do this you can start looking at the patterns produced and start theorizing and start writing conjectures. But all that you discover is not related to the place on the number line. It is related to the nature of the extra machinery.
     
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  12. Jul 29, 2007 #37
    a race


    Basically, I meant to say that you "start" the counting system. You also "start" to count USING the Peano system. Now, for each step, there will be a result for each system. Let's say that there is a set of results for the counting system and a set of results for the Peano system WHEN USED as a counting system. Now, just map the two systems formally with these results in mind. If this could be done, then I guess you can say the two systems are equivalent in the sense of those sets of results. However, you can not impose the notion of prime from the Peano set of results back over to the counting system. That is all I am wanting to do. And I want to know what it means for the notion of primality. Just trying to open up discussion about all this in layman's terms.
     
  13. Jul 29, 2007 #38

    CRGreathouse

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    Of course some could argue -- and I think I would -- that this extra machinery is the number line, not the counting axiom schema. So far that's not even strong enough to tell us that "2" comes after "1".
     
  14. Jul 29, 2007 #39
    I don't think there was too much extra machinery. I just replaced his notation (da da da <end> for the number 3) for another more easy to type (da da DA for 3), and defined addition as the concatenation of sequences, which is only natural when counting "da da da".

    The prime definition, on the old notation, would not change the concept. It would say, "a prime is a number where the first 'da<end>' do not coincide with any 'da<end>' for smaller numbers". I merely replaced 'da<end>' by 'DA', and defined addition. The fact that you don't explicitly mention the <end> at the end does not make the concept of 'end' disappear.
     
  15. Jul 29, 2007 #40
    strong enough


    So, if the extra machinery is the number line, then Peano might possibly might not have been biased by an intuitive notion of a number line? Given Peano axioms (we just happen to be using Peano axioms for sake of discussion) what do they do (in context of discussion)? Do they:

    A) create the number line?
    -- or --
    B) create the facility to "talk" about the number line?
     
  16. Jul 29, 2007 #41
    primes: what is it?

    Are you saying that you are in agreement that the notion of prime is not due to a strict position on the "number line" (whatever the hell a number line is) and instead it is due to structure of the 'meta-data" you are adding when you change the appearance from "da" to "DA"?
     
  17. Jul 29, 2007 #42

    CRGreathouse

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    I think that once you put connections in between the numbers (so that "3" is right between "2" and "4", where in your axiom schema right now there's nothing special connecting the three) you can form the concept of primes.
     
  18. Jul 29, 2007 #43
    positions

    Actually I do not want to be able to have the concept of primes. That is why I left it as a counting system. Leaving it that way, I want to map the bare counting system to the Peano system's version of the counting system. After all, I am assuming that the Peano system can indeed "simulate" the bare counting system. Is this possible? Yes I think it is. Why do it? For the sake of understanding the notion of "prime" separate from the notion of the position of the "item" on the "number line."
     
  19. Jul 29, 2007 #44

    CRGreathouse

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    But your line doesn't have position right now. "7" is just as close to "1048576000000000" as it is to "6".
     
  20. Jul 29, 2007 #45
    Algorithmically speaking..... I can just as easily interpret the Peano axioms algorithmically since Peano successor function is very "algorithmic." Yes my counting system does not have position in terms that you speak of. It only has algorithmic position. Can we map this notion of algorithmic position (or step of execution - comp. sci. terms) to the "positional" stuff that we get out of the Peano axioms?
     
  21. Jul 29, 2007 #46

    CRGreathouse

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    What do you mean by algorithmic position?
     
  22. Jul 29, 2007 #47
    mapping retake

    I was referring to the positional "stuff" that you get out of the Peano system.

    Let try this: Just ignore positional stuff in the Peano system and try to get it to produce what we are calling a "counting system." Let us say the Peano system can do many things. One of the things it can do (we hope) is just simulate the basic counting system. we have been talking about. Now, equate these two systems Once you map them, then allow the positional stuff to come back into view on the Peano side. With it comes the notion of prime but you can not impose that notion of prime back onto the basic counting system that was mapped to the peano counting system.
     
    Last edited: Jul 29, 2007
  23. Jul 29, 2007 #48
    steps

    Steps. How does the peano successor function produce the successor? In zero time? Does the framework allow one to talk about the successor function in terms of "steps." How does the successor function "compute" the successor of x? Is it a magical filter that pumps out numbers but does not let you look into it ?

    Obviously, time is not a factor in the "ether of mathematics and abstractness" but what is preventing me from saying there are 2 steps from S(4) to S(6) ?
     
  24. Jul 29, 2007 #49
    successor


    Actually, I think about it more and I am convinced that the Peano system definately allows you to look at what is happening in terms of algorithm. Algorithm implies steps. Perhaps Peano wanted an algorithmic viewpoint, I don't know.
     
  25. Jul 29, 2007 #50
    Time for some side question:

    Is there a way to map one axiomatic system to another? Has it been done for systems that are similar? Can it be done from one simple system to a system that is more feature rich but which still simulates the simpler system?

    What exactly does it mean to map a formal axiomatic system to SOMETHING ELSE. Is this the technique employed by Godel in his most excellent work?
     
    Last edited: Jul 29, 2007
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