wonderings

the multiplicative group over the positive rational numbers is just the direct group product of infinitely many multiplicative groups of powers of some (essentially arbitrary) prime. until you include addition, no one prime can be said to be bigger than another, the "value" of each prime is completely arbitrary/even meaningless, and you can add or remove primes freely without inducing holes.

for example:

the powers of two under multiplication/division are a closed group -- if you multiply any one of them by any other you will end up with another power of two. so, since we always know our number will look like 2^x, let's just drop the 2^, and write (x). now, (x) * (y) = 2^x * 2^y = 2^(x + y) = (x + y). so we can redefine the product operation (x) * (y) as just (x+y).

but wait a minute. if all we have now is the operation (x) * (y) = (x + y), there's absolutely nothing in there forcing the power at the bottom to be two. it might as well be "apples", for all we know or care.

and, as an experiment, what happens when we let every number have two values, instead of just one? let's define (x, y) * (z, w) = (x + z, y + w). what we have now is a multiplication system with two primes. we've come up with a way to represent all composite numbers with these two primes as their only prime factors, as a function of the number of each prime factor in a number. but we don't know what factors are. maybe (x, y) = 2^x * 3^y, but it could just as easily be apples and oranges because we don't actually know what those two primes are, and the two columns never interact in any way, it's meaningless to talk about which is "bigger", and how they are ordered within the number system.

in conclusion:

all the weirdness with primes isn't really anything to do with primes themselves, it only comes about when you combine addition with an infinite-prime multiplicative group. and interestingly, the rules for addition seem very similar to the rules for a single-prime multiplication group.