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#19
Jul2807, 07:34 PM

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* 1 is a natural number
* For all x, Sx is a natural number * For all x, Sx is not 1 * For all x and y, x = y iff Sx = Sy Okay. So regarding the obscure number line that sort of exists before Peano touches the paper with his pencil, I imagine what would happen if Peano wrote the following: * 32654 is a natural number * For all x, Sx is a natural number * For all x, Sx is not 32654 * For all x and y, x = y iff Sx = Sy It would be totally cool with me. But it sort of points out that the number line is still there regardless of whether Peano writes the axioms down or not. Sure I take your point that it is a rather platonistic statement but I can not seperate out the platonisticism when talking about this stuff at this level. In fact, just for fun, I will add a few more: * 99 is a natural number * For all x, Sx is a natural number * For all x, Sx is not 99 * For all x and y, x = y iff Sx = Sy * 10010001 is a natural number * For all x, Sx is a natural number * For all x, Sx is not 10010001 * For all x and y, x = y iff Sx = Sy * 666 is a natural number * For all x, Sx is a natural number * For all x, Sx is not 666 * For all x and y, x = y iff Sx = Sy (the above I could not resist!) 


#20
Jul2807, 07:42 PM

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"1" is just a label, it could be anything. Consider this model of Peano arithmetic:
"1" is a triangle. If x is a polygon, Sx is a polygon with one more side than x; otherwise, Sx is a pink unicorn. x = y iff x and y are polygons and x and y have the same number of sides. For example S(S(1)) (that is, "3") is a pentagon. This works perfectly well  all the Peano axioms can be made to hold in this system, even though the underlying objects are not "numbers" in any normal sense of the word. If we went through all the usual definition we would find that x + y would be defined as a polygon with two fewer sides than the total number of sides in x and y. 


#21
Jul2807, 07:53 PM

P: 120

Yes I am in agreement with you about the labels. But, I am not convinced that Peano wrote that stuff without already having been biased by the notion of the number line.
Also, given * 1 is a natural number * For all x, Sx is a natural number * For all x, Sx is not 1 * For all x and y, x = y iff Sx = Sy I still maintain the early phrase "For all x" is unfortunate. It clearly can be interpreted as saying "for all the x that already exist." This is very much a problem in my opinion. It feels as if the system is riding on the edge of chicken and egg. I am sorry but it will take me some time to absorb that it is not a chicken and egg system (I will take the mathematicians word for it for now). If, however, the numbers can exist without the successor function definition (example: in terms of counting only and not in terms of talking about where the counting stops) then clearly there is a problem with the definition of Prime. It refers not to the "position" where the counting stops UNTIL the operational axioms are applied. 


#22
Jul2807, 08:04 PM

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#23
Jul2807, 08:30 PM

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Let us assume that there can be an axiomatic system which does not allow you to construct the natural numbers. Assume also that the axiomatic system just states all the wouldbe numbers in terms of "steps" in a counting process.
Obviously, why do this since the point of the axiomatic system is to be able to build numbers. Don't worry about that for now. So here is the "foo" axiomatic system: "foo" axioms: "foo" "foo,foo" "foo,foo,foo" "foo,foo,foo,foo" "foo,foo,foo,foo,..." That was my best attempt at defining a simple basic axiomatic system which just states the steps at each count of a counting process. Obviously, to talk about each step in terms of a number one needs to have some structure about what it is that you are talking about. Once you add the structure you can finally say that "foo,foo,foo,foo,foo" is really 5. But until you have done so, the "foo,foo,foo,foo,foo" is not a prime even though it falls in the same position on the number line as the 5 from the other axiomatic system. It is not a formal example but I am trying my best to get to that point. Any ideas how we can formally write an axiomatic system which just lists the "foo" ?? Perhaps we can call it the "counting axioms." 


#24
Jul2807, 08:38 PM

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Axiom 1: 1 is a natural number. Axiom 2: 2 is a natural number. . . . OK, so now you have a system where you cannot add or take the successor, but you have the natural numbers. (You can call them "foo, foo" and the like if you wish, but names are just labels to mathematicians so I'll just call them this for now.) So in the context of this system containing your counting axiom schema, what is the definition of "prime"? Or is your point that you can't even define it here? (Mathematicians would say, informally, that they don't have the 'machinery' they need.) 


#25
Jul2807, 09:00 PM

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Yes. It is my point. In that system there is no way to define prime. Hence, primality is not a feature until more machinery is added. 


#26
Jul2807, 09:29 PM

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From the point of view of formal logic, having a multiplication operation makes a rather large difference.
If we use firstorder 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 (firstorder) 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 higherorder logic. 


#27
Jul2807, 09:43 PM

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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. 


#28
Jul2807, 10:27 PM

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[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] 


#29
Jul2807, 10:55 PM

P: 120

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. 


#30
Jul2807, 11:03 PM

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"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) 


#31
Jul2807, 11:07 PM

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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. 


#32
Jul2807, 11:12 PM

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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 multibranch 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. 


#33
Jul2907, 01:07 AM

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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 reedition of the Sieve of Eratosthenes.) 


#34
Jul2907, 09:15 AM

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That aside, I'm still not sure I quite follow. What is the motivation behind the branch terminology? 


#35
Jul2907, 12:30 PM

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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.



#36
Jul2907, 01:44 PM

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