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November 2nd, 2012, 02:21 PM   #1
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The paradox between prime numbers and natural numbers.

Prime numbers are the building blocks of natural numbers. But how can you identify a prime number without using natural numbers?

For instance you can identify 7... as 7. The other means is identifying't it as the 4th prime. But in both cases you use natural numbers to define prime numbers. Either in cardinal terms or ordinal terms you can't avoid using natural numbers to construct a prime number,

It looks like prime numbers construct the natural numbers and natural numbers construct the primes.

This seems like a cyclical paradox. How can you resolve it?

One way is to announce the unit as the building block and I'm not sure how I'd describe the primes.
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November 2nd, 2012, 05:11 PM   #2
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Re: The paradox between prime numbers and natural numbers.

Quote:
Originally Posted by Eureka
Prime numbers are the building blocks of natural numbers. But how can you identify a prime number without using natural numbers?
I would not build the natural numbers by using prime numbers, I would just use Peano's axioms.

Even when you do try to create the natural numbers with the primes you still have a severe problem. You need to have a new operation , since our old operations are defined by using the natural numbers.
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November 2nd, 2012, 05:18 PM   #3
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Re: The paradox between prime numbers and natural numbers.

It looks like semantic confusion.
1) Primes are natural numbers themselves.
2) Primes construct other natural numbers but cannot be constructed by definition. They can only be defined, as you mention, by other natural numbers (including other primes)
Here's one of possible systems of definition of odd numbers based on the formulas
N=3a+1; N=3b+2.
Where a?2 even number; b?1 odd number.
The system includes not all odd numbers but all primes. All natural numbers are involved in definition.
3?1+2=5;
3?2+1=7;
3?3+2=11;
3?4+1=13;
3?5+2=17;
3?6+1=19;
3?7+2=23;
3?8+1=25;
3?9+2=29;
3?10+1=31;
3?11+2=35;
3?12+1=37;
.............
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November 3rd, 2012, 03:46 AM   #4
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Re: The paradox between prime numbers and natural numbers.

It is tempting and in some ways useful to think of primes as atoms and composites as molecules made up of their prime factors or employ some similar metaphor. But, as Greenlizard0 pointed out, those who have attempted to derive natural numbers from something more basic have not gone down that road at all in their axiomatizations, but instead take 0 as given and define a "successor of" operation, so that 1 is the successor of 0, 2 is the successor of the successor of 0, etc. This method of course apes addition by 1 and while primes are special with respect to multiplication and division, they are just a bunch of other numbers with no particularly special properties with respect to addition (and subtraction).

I would add, however, that the more you study logic, the more cases you'll find where it seems that you are already allowed to know in the metalanguage the meaning of things strenuously defined in the object language.
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November 3rd, 2012, 04:51 AM   #5
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Math Focus: primes of course
Re: The paradox between prime numbers and natural numbers.

okay, I'll add my take
I would start by conceding 0,1 as unto themselves. We then need add/sub, and denoting adjacency by 1 using the suffix "N","P" . We use multiplication only for clarity of equivalences, it is not needed.
We will label from "1" denoting the first prime. I will show substitutions, but they muddle things quickly, just realize every other number can now be expressed/derived using only the number 2 (my "1") and the math operations stated. Realize the numbers are not generated in the normal counting order by necessity.
2 "1"
3 "2"= 1P
6 1*2 = 1 *1P or 1+1+1
4 1+1 (composite whenever xpressible via pure addition of another(single) core number)
5 "3"= (1 * 1P)N = ( 1+1+1)N
7 "4"= (1 *1P)P= ( 1+1+1)P
8 1+1+1+1
9 2+2+2 = 1P+1P+1P
10 1*3 = 1* (1 * 1P)N or 1+1+1+1+1
12 1*1*2 = 1*1*1P or 1+1+1+1+1+1
11 "5" = (1*1*1P)N = (1+1+1+1+1+1)N
13 "6" = (1*1*1P)P= (1+1+1+1+1+1)P
etc etc
17 1*1*1P+(1 * 1P)N = 1+1+1+1+1+1+ ( 1+1+1)N

[hopefully no typos]. Of course this presumes the A129912 conjecture.
The post will mess with my format spacing, sorry
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