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Whole prime number

  1. Jul 30, 2007 #101
    I think we take the machine analogy too far. Axiomatic systems are meant to provide a set of initial assumptions from which to construct the rest of the building. They are not algorithms, but logical propositions accepted by convention as true (so that all derived statements can be proven true). By themselves, they do not travel in time; our reasonings and explanations do, but only because our talking does.
    I'm getting lost. I thought you said you didn't want the ticks to represent numbers, since there is no way of telling the 5th from the 625th.
     
    Last edited: Jul 30, 2007
  2. Jul 30, 2007 #102
    order and elements

    Yes you are following me. I felt originally that if a system has the two things (elements and order) you could not guarantee the ability to talk of "primality."

    But now I am side tracking to a system where order is out the window. I just want a ticking system (unforntunatley, the algorithmic interpretation of step, next step, next step, next step, etc. DOES indeed imply order.
     
  3. Jul 30, 2007 #103
    internal state

    Sorry, that term was NOT brought into the discussion by me!
     
  4. Jul 30, 2007 #104

    CRGreathouse

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    So you want a countably infinite number of elements, but no comparisons between them. Seems like the axiom schema "For each natural number n, there is an element distinct from at least n others" of your counting scheme is what you want. Of course I can't think of a way to use that at all -- not for finding/defining primes, not for counting sheep, not for anything. It's essentially identity calculus.
     
  5. Jul 30, 2007 #105
    stop using

    Sure I did not expect that anyone would want to use this system. Given what you just said about the identity calculus my question is can the Peano axiom system be built from the identity calculus?

    Lets just say I am interested in decomposing the Peano axioms into building blocks (or feature sets) much like you would decompose some number into a combination of primes.
     
  6. Jul 30, 2007 #106

    D H

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    With this system you need an axiom for each number. This is way too much effort. Moreover, how do you define addition? Why does 2+3=5? I don't see that relation anywhere in your system. The definition of addition falls out naturally from the Peano axioms. Multiplication and division fall out naturally from the definition of addition.

    In comparison, defining the "numbers"1 recursively (or inductively) requires but three axioms: an axiom stating that "one"2 is a "number", another that no number has "one" as a successor, and a third stating that if x is a "number" then the successor of x is a "number". Recursion/induction is central to mathematics. It is extremely powerful.

    Notes:
    1The Peano axioms define the "natural numbers". Using any other naming scheme in conjunction with the Peano axioms generates a set that is isomorphic (identical characteristics and identical behavior) to the natural numbers.

    2Modern treatments start with zero rather than one so that addition and multiplication can be easily defined based on the Peano axioms.
     
  7. Jul 30, 2007 #107

    CRGreathouse

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    In what sense? Sure, by adding axioms, but in that sense you can build the Peano axioms from nothing -- so one way to build the Peano axioms from identity calculus is to ignore it and add all the normal axioms.

    When I say identity calculus, I mean a system with no operation except "=". If you were somehow able to count the number of cows you and I had (the system has no way to count, but if you were given the numbers by proposition or axiom) then you could say that the two were different numbers but no more.
     
  8. Jul 30, 2007 #108
    axiomatic systems

    Okay I admit I just threw those numbers out there when I said:
    x 1
    x 2
    x 3
    x 4
    x 5

    Why do you care that the system has "too much effort"? It is still a valid axiomatic system. I am trying to learn about the process of writing axiomatic systems. I am just trying to learn about the different things I can create with an axiomatic system. Can I create an axiomatic system with infinite axioms? Not in practice but in theory. That is at the far extreme edge of what kind of systems you can create but it is still worth study.

    I must ask you then, to define a prime number in terms of addition (using the Peano axiom system). It should be easy for you to do since multiplication and division just "fall out naturally from the definition of addition."


    So I have another question: What do you call the process of writing/defining an axiomatic system? For now I will just call it "FORMALIZER 1.0" since I do not know what it is called but I want to pose another question specifically about it:

    Is "FORMALIZER 1.0" built using recursion/induction?
     
  9. Jul 30, 2007 #109

    D H

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    Addition: Seeding the Peano axioms with zero (rather than one), define
    • a+0 = a
    • a+S(b) = S(a+b) for all a,b in N

    Multiplication is similar: Define
    • a*0 = a
    • a*S(b) = a*b+a for all a,b in N

    Like I told you, recursion is extremely powerful.
     
  10. Jul 30, 2007 #110
    cool

    Very cool looking. You just defined addition and multiplication. But how do you define a prime number with addition only?
     
  11. Jul 31, 2007 #111

    CRGreathouse

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    Nice. I'd use different units for addition and multiplication, though. :tongue:
     
  12. Jul 31, 2007 #112

    CRGreathouse

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    We've done it already -- there can't be a solution to x + x = y, x + x + x = y, ..., (x-1)y = x. Alternately, define other operations recursively and use them to define it more traditionally.
     
  13. Jul 31, 2007 #113
    without addition

    Okay, now this is getting interesting. I need to study the recursive versions a little while. However, I am still left with an important question. If addition is not defined then can you still get numbers? I think someone said earlier that addition is given by default in the Peano system somehow due to the successor function. More generally, can I, using the axiomatic method, define the natural numbers without defining addition?

    If yes, then the notion of "prime" is due to the addition or other operations and not the actual number as it lies on the number line. I hope this question makes sense.

    Also, Can we talk about "prime" in terms of the metronome system? My guess is "NO". This is interesting because, in my opinion, the number line and the "tick" line of the metronome system are the same thing or same "form".
     
  14. Jul 31, 2007 #114

    CRGreathouse

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    What's a natural number? Certainly you can define things without defining addition, but could they be considered natural numbers without successors or the ability to add? Once again, philosophy not math. If you have a definition in mind it becomes math again.

    I'm not uncomfortable with philosophy, but I know even less of it than I know of math -- I took only a few philosophy courses in college, though I did well in them.

    I don't understand your use of the term "metronome system".
     
  15. Jul 31, 2007 #115
    addition is what?

    So from your point of view, addition is nested succession? Perhaps you could say that addition is a way to specify a "short-cut" style of succession?


    Sorry. Earlier I attempted to switch from "counting system" to "metronome system."
     
  16. Jul 31, 2007 #116

    CRGreathouse

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  17. Jul 31, 2007 #117
    short cutting


    Thanks for the extra information. You say you do not think of addition as a shortcut and I can accept your viewpoint. Just for the sake of discussion, don't you think that having defined "addition" is essentially the reason why you do not have to rely upon the "counting" or "metronome" interpretation of what comes out of the Peano system?

    Also, for the Peano axioms, surely there is some kind of "counting" or "metronome" feature. The reason I reiterate this idea is because, if you think about it, the successor function is always "counting" or "ticking" from the 1. Multiplication, addition, in this peano system, is always given in terms of a count from the 1 mark. Essentially, there is no short cut in a recursive system because everything is in terms of the successor function (in relation to 1). So, addition is inherent. Fine. But as soon as you "reach" into the system and "tag" something as "addition" then you have created a meta-shortcut which is quite useful for interacting with the system. Addition is a "user" short cut not a system short cut.
     
  18. Jul 31, 2007 #118

    CRGreathouse

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    OK.

    I agree that the system can be seen as using the successor operation underneath, and that's usually the way things are defined. I don't see it as a shortcut at all on the 'user' level, though: as I mentioned the full operation on rational/real/complex/etc. numbers doesn't follow from the successor operator and must be defined differently.
     
  19. Jul 31, 2007 #119
    defining numbers


    If the operations are not defined then basically you just have numbers that are defined in terms of their positions in relation to "1"??? Would I be correct in saying this?

    If this is the case then, once again, I don't see how a number can be "prime" without definitions of the operations (addition, multiplication, etc.). We have agreed that we can define the numbers with just the successor function and that the numbers are fully defined despite not having operations. I just can't understand how anyone could look at this system and say that some particular number is "prime."
     
    Last edited: Jul 31, 2007
  20. Jul 31, 2007 #120

    CRGreathouse

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    Not in general, no. It depends on what is given. Unless you're more specific on what is defined rather than what is not, there's not much I can say.

    One simple way would be to define primality, or build it directly into one's system. A natural way (to me) to define numbers would be to start with the primes as atoms and define the positive integers as the product of some collection of primes, with equality if and only if the number of each prime was the same. The natural "successor" operation S_p(n) would then increment the count of a single prime by one, i.e. "multiply" the number by that prime. Addition would be a complex relation that would be shown to always have a unique answer only by a theorem as profound as the fundamental theorem of arithmetic is on our system.
     
  21. Jul 31, 2007 #121
    makes sense

    I think I am getting close to understanding my own confusion.

    I still have a few misunderstandings. When Peano fully defined the successor function, did addition fall out automatically (I think this was a point in an earlier posting about recursion)? Looking at the axioms on wikipedia, I can't see an explicit definition of addition. Interestingly the wikipedia editor for the Peano axiom topic has written the following:
    From that statement, it seems as though the addition is indeed built into the successor function.

    I think I finally understand the difference between the Peano axiomatic system and the counting system we talked about earlier. I propose the following thought experiment:

    All the arithmetic operations of the Peano system (and hence the notion of prime) could not exist if there was not a reference "point" defined on the "number line." If you take that first axiom away from the Peano system then all you have is a system that acts like a "metronome" ( the "counting system" that we have been talking about).
     
  22. Jul 31, 2007 #122

    CRGreathouse

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    Off the top of my head:

    x + 0 = x
    x + S(y) = S(x + y)

    These two allow any two numbers to be added, since you just decrease one step by step until it is zero, increasing the sum likewise.

    That was just terminology. The successor just picks the next number; addition is defined in terms of it. It's convenient to mention that once addition is defined x + 1 will be the same as S(x), but that just falls out of the above definition since S(0) = 1.

    Which is the first axiom?
     
  23. Jul 31, 2007 #123
    oops

    Ooops! Indeed, the axioms are not ordered! I mean the axiom which states: "1 is a natural number" Also, we might be looking at two different versions of the axioms. I am looking at the list on the wikipedia (which are slightly rephrased from the original perhaps). Anyway, take that out and you basically sever addition and multiplication. Perhaps you also have no way to say what a number is. However, the number line is still there. The form has not been changed, destroyed or altered in any way (this will be very profound to me if it is indeed true).
     
    Last edited: Jul 31, 2007
  24. Aug 1, 2007 #124

    CRGreathouse

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    Without the axiom "1 is a natural number" (which in modern terms is "0 is a natural number") you can't prove the existence of numbers at all.
     
  25. Aug 1, 2007 #125
    Actually that number 1 is not an arbitrary reference point. In informal terms, it is "the step of the successor": when defining addition, we started by a + 1 = S(a). Similarly, when extending the axioms with S(0)=1, the number 0 turns out to be the neutral element of addition, a + 0 = a, all as a consequence of the initial axioms and the definition of addition.

    More formally, if you define a function f: N* x N* -> N* (N* being the numbers including 0) as
    d(a,b) = the number 's' such that, for a <= b, we have a + s = b;
    and for a > b, the number d(b,a)​
    then I'd say the function d(a,b) passes the requisites to be considered a metric (non-negativity, identity of elements with distance 0, symmetry and triangular identity), so that N* plus the function d() is now a metric space. So now we can speak of distance: and the distance d(a,S(a)) is exactly 1.
     
    Last edited: Aug 1, 2007
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