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.
 A: You say "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 $2a+b=1$". This deduction seems hasty to me. I assume you're thinking "two natural numbers that are relatively prime have a sum that has no prime factors in common with either; and the only natural number that has no prime factors is 1". However, you're assuming to start with that $2a+b$ is a natural (or perhaps supernatural) number, but there's no reason that $2a+b$ has to be well-defined.
In fact, your claim together with mjqxxxx's modification could probably be combined to give a proof that addition cannot be defined on the supernatural numbers.
A: 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.
A: It is possible to embed the supernatural numbers in the hypernatural numbers in a somewhat natural way. Indeed, let $\mathbb{U}$ be a free ultrafilter on $\mathbb{N}$, and let $^* \mathbb{N} = \mathbb{N}^\mathbb{N} / \mathbb{U}$ be the hypernatural numbers. We can identify the supernatural number $p_1^{\alpha_1} p_2^{\alpha_2} \ldots$ with the hypernatural number containing the sequence $(p_1^{\alpha_1}, p_1^{\alpha_1} p_2^{\alpha_2}, \ldots)$ if $\alpha_1, \alpha_2, \ldots$ are finite. There are multiple ways to raise $p$ to an infinite power in the hypernaturals, but it would be possible to choose one way and then stick to it. The hypernatural numbers form a (very nicely behaved) ring--however, if we embed the supernatural numbers in them in the manner described above, they are not closed under addition. When you add supernatural numbers, are you okay with getting something else?
