Defining addition of supernatural numbers? In the comments on this question Bill Dubuque mentions the supernatural numbers.   My curiosity was piqued by the statement on Wikipedia that "there is no natural way to add supernatural numbers" and I soon invented this example:
Let $a$ be the supernatural product of all primes congruent to 1 mod 4, and let $b$ be the supernatural product of all primes congruent to 3 mod 4.  Because GCD is defined for supernatural numbers, and the sum of two relatively prime numbers is relatively prime to each of them, we can say that $2a + b = 1$ and also that $a + 2b = 1$; adding these gives $3a + 3b = 2$ or $a + b = \frac{2}{3}$.  The value $\frac{2}{3}$ can apparently be interpreted as a "super-rational" number, a supernatural-like number where negative exponents are permitted.  So it seems  that I can give a consistent definition of addition at least for some supernatural numbers (although the result in this case is "super-rational").
What is the basis of the claim that "there is no natural way to add supernatural numbers"?  Do the assumptions underlying my idea lead to any contradiction?  If not, to what extent can it be extended to allow the addition of more general forms?
EDIT: I hadn't read the article closely enough to realize that supernatural numbers are allowed to have exponent values of $\infty$, and also it has been pointed out that my idea does not work in any case.  What remains of this question I feel is too unfocused.  I am accepting Greg Martin's answer.

Addition cannot be defined on the supernatural numbers. Indeed, assume the contrary. Then we get $$2^{\infty}\neq 3\times 2^{\infty}=2^{\infty}+2^{\infty}+2^{\infty}=(2^{\infty}+2^{\infty})+2^{\infty}=
2\times2^{\infty}+2^{\infty}=2^{\infty}+2^{\infty}=2\times2^{\infty}=2^{\infty},$$
which is the contradiction.
