Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to $35 to read it.

So, I get it. My brain is more wired for analysis than algebra; so this isn't the first time that linear algebra has been a useful bridge for me. I see that we could have a 'vector space' of infinite-dimensional vectors where each vector (a1, a2, ..., an, ...) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)... and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.

I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of 'vector space' to make certain things line up. But I can already speculate on how the choice of the "scalar field" for specifying the a_i would have interesting consequences:

• non-negative integer 'scalar field' --> the positive integers,
• all integers 'scalar field' --> positive rational numbers,
• complex integers --> finally include the negative rationals.

I'd like to read more. What sub-field of mathematics is this?