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Whole prime number

  1. Jul 30, 2007 #81
    While you create new systems, try to add to them a very desirable property than Peano's have: the ability to express very big (or infinite) objects in a finite, even very compact, space of symbols. This is what makes better a system like "1 is a number, Sa is a number if a is" than "foo is a number, foo foo is a number, foo foo foo is a number, ...", or than "1 is a number, S1 is a number, SS1 is a number, SSS1 is a number, ...".
    Last edited: Jul 30, 2007
  2. Jul 30, 2007 #82


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    My basic answer would be when you make a system strong enough to support a discrete chain/total order, you essentially have a way to talk about divisibility and thus primality.
  3. Jul 30, 2007 #83
    I have some confusion here. The relation < defines a total order on R, yet that doesn't make R isomorphic to N. Without that isomorphism, you get 7 is not a prime because it is divisible by 7/2.
  4. Jul 30, 2007 #84
    no features please

    I am not interested in building a system with "very desirable property than Peano's have." I have a specific reason why I am limiting the functionality of the counting system.
  5. Jul 30, 2007 #85
    No problem; as long as you don't count over 100, you won't spend too much paper.

    But CR has a point up there. Divisibility is actually a very basic concept, that can come from addition, not necessarily multiplication. For instance, define numbers with dots, x, xx, xxx, xxxx..., and addition as concatenation. Then define divisibility by these two axioms:
    1) Every number divides itself, f.i. xxx divides xxx.
    2) If b divides a, then b divides a+b. That is, if xx divides xx, then xx divides xxxx, and also xx divides xxxxxx ...
  6. Jul 30, 2007 #86
    talking about prime

    My basic thought is that "talking about" does not mean it is a "hard asset" of the system. For example, the Peano axioms lend themselves quiet well to "algorithmic" discussions due to the successor function- it lends it self quite well to a mechanical "stepping" system. However, that doesn't mean there is any real ability for the system to "step around" on the number line. It could very well just as easily magically make the numbers "poof" into existence (since there is no notion of time constraint ).

    Anyway, my question about your response is: what system is weaker than a discrete chain/total order? (also, I am not totally sure what you mean by discrete chain/total order but I have a good guess that it is something that just "ticks").
  7. Jul 30, 2007 #87

    Okay I see the problem. I want a system that just "ticks." Your system is nested ticking. In my system I dont want to do anything except be able to produce the next "tick". There is no notion of nested ticking in my system. How can this be formalized with an axiomatic system? I will just call it a ticking system. Is such a ticking system the most basic kind of axiomatic system with the least amount of feature? In such a system I think divisibility is not definable. Basically, you push off the job and definition of divisibility to the observer or user.
  8. Jul 30, 2007 #88

    Exactly the point I made earlier. It also helps if you do not eat very much MSG (monosodium glutamate) since it is an excitotoxin and directly attacks cells in the short term memory area of the brain making it hard to count in unary...
  9. Jul 30, 2007 #89
    axiomatic nesting

    Not to get off topic but I must ask because I am so curious: Is the Peano system inherently nested? Does it have built-in nesting? Do all axiomatic systems have built-in nesting??? Is nesting just coming about because of the way the system is being "used" by the "user"?
  10. Jul 30, 2007 #90
    I'm not sure of what you mean by "nested". I think you mean,
    x, xx, xxx, xxxx... are numbers​
    is "not nested", while
    x is a number; also, if A is a number then Ax is a number​
    is "nested". I think most people here would say both are one and the same.
  11. Jul 30, 2007 #91

    But if the system is viewed as an algorithmic process, then how do you distinguish? Especially if we are talking about a system that only can only tick. How can we limit the expressiveness of an axiomatic system so that all you can do is "poke" it so that it "ticks". Can we have a one-to-one input/output system. Axiom systems like Peano have many ways to "input" your "statements" to make them "produce" an output. I do not know how the formal mathemticians talk about the "usage" of the axiomatic systems at this level of abstraction, but I see it with the input/output metaphor.

    Nested counting is where at each step of the count, the process starts again from "one."

    x 1
    xx 1,2
    xxx 1,2,3
    xxxx 1,2,3,4
    xxxxx 1,2,3,4,5

    I am saying why waste so much effort? Just do this:
    x 1
    x 2
    x 3
    x 4
    x 5

    I had to put the numbers in there for visualization but I am saying that I just want a ticking system.

    Anyway, Here is my focus:
    I want to define a ticking system using the axiomatic method. But, I do not want the system to do anything except tick! No nesting. Can this be done with the axiomatic system or is it too flexible at it's core such that it can not make such limitations? This is a short side study on the nature of axiomatic systems.
  12. Jul 30, 2007 #92
    In this system, what differentiates 5 from 2? Other than the fact that they were created by different axioms? This system has no notion of order, unless you explicitly put it in, in which case you get nesting as you put it.
  13. Jul 30, 2007 #93
    In other words, this ticking machine seems to have no internal state. When it ticks and says 'x', there is no way of telling if it is the first 'x', the 5th or the 625th. On the other hand, if it *does* have a state, then you should consider how the state is represented, and call this representation a 'number'.
  14. Jul 30, 2007 #94

    Yes. It is my point. You cannot differentiate those numbers. I just put them in the post for sake of moving forth in the discussion. If you view the system as an algorithmic process, then it kind of does have order. Okay. So are you saying that using a formal axiomatic theory, I can not create a ticking system with order AND which does not provide notions of divisibility, prime, and anything else related to numbers or number bases? I am interested in the answer which is why all this time I am not concerned about cool stuff like addition or division.
  15. Jul 30, 2007 #95

    You are correct in saying you can not figure out if it is the first 'x' or the 5th 'x' or the 625th 'x'. I said already that I want the user of the system to worry about that. I don't want the user of the system to expect that the system to tell them via some particular feature of the system. They just use the ticking system like a metronome (a metaphor obviously).

    Let me put it this way- can there be an axiomatic version of a metronome?
    Last edited: Jul 30, 2007
  16. Jul 30, 2007 #96
    internal state

    Do axiomatic systems like Peano have internal state? Can some axiomatic systems have an internal state and others not have one?
  17. Jul 30, 2007 #97
    I fail to see how a system with no numbers fits in a number theory forum, but for the sake of the discussion let me provide an example. The boolean operator 'not' behaves as you want. Or, if you prefer, a function f(x) = 1 - x. But unless we begin counting time, or sequence steps, the only thing we produce is the set {0,1}, with no additional structure, operations or functionality.
  18. Jul 30, 2007 #98


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    You want your system to have elements ('number' which function as atoms or ur-elements) and a (presumably transitive) order on those elements, and you want to know if all systems with those properties can talk about primality in some restricted way or not? Is that right?
  19. Jul 30, 2007 #99
    building blocks

    The reason is not sake of discussion. The reason is of great importance. One of the points all all this discussion is that you have built on top of something to get "numbers". The thing (stepping stone) you start with is the tick system. So, whether or not you agree with what I am doing, I really need help trying to formalize the tick system without using something that already has notions of "number". Unfortunately, that math education imposed upon us does not even begin to get into these concepts.

    Look at the peano system as one holistic system or look at the Peano system in terms of modular building blocks. If you take the modular building block approach then I am saying that before you can construct numbers you have to build on top of the tick system. Therefore, it has lots to do with the topic of Number Theory.
  20. Jul 30, 2007 #100


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    Internal state? :confused:
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