Defining addition of supernatural numbers?

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.

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.