Quote:
Originally Posted by Don Blazys
For example, in number theory, a "perfect number" is defined as:
"A number such that the sum of its proper factors is equal to itself".
Thus, if we allow both "multiplication by unity" and "multiplication of unity",
then 6 can be factored as:
6=3*2*1*1, where 3+2+1+1=7,
and the entire concept of a "perfect number" simply collapses!
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Why would the number 1 want to be included in the divisors twice? The divisors of 6 are {-6,-3,-2,-1,1,2,3,6}. A perfect number is a positive integer which is the sum of its proper positive divisorsproper meaning to exclude the number itself. The proper positive divisors of 6 are then {1,2,3}.
I think including 1 twice would assume the divisors are being multiplied, which isn't the case. That 1 2 3 = 6 (or 1 1 2 3 = 6) is coincidental as far as perfect numbers go. The next perfect number, 28, sums perfectly (28 = 1 + 2 + 4 + 7 + 14), but doesn't multiply perfectly (28
1 2 4 7 14).
So, I'm confused what you did there. And, also in the case of the fundamental theorem of arithmetic, an example such as
doesn't include the factor 1 because 1 isn't prime. My only guess is that you're thinking
wouldn't be unique if
were allowed. Is that the case? I'm probably way off base.
In any case, the multiplicative identity follows as second order logic from the Peano axioms of the natural numbers. So, I don't see how it is at all incompatible with number theory. The wikipedia page on
Peano axioms under the arithmetic section has:
Quote:
It is easy to see that 1 is the multiplicative identity:
a 1 = a (S(0)) = a + (a 0) = a + 0 = a
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and using more common logic syntax, axiom #7 under Equivalent axiomatizations:
Quote:
7.  i.e., one is the identity element for multiplication.
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My understanding is that one is included in the natural numbers, so the above explicitly allows 1 x 1 = 1... which I can't reconcile with your statement about number theory not allowing "multiplication by unity and multiplication of unity".
~modest
