I would be extremely grateful for any help in understanding this question concerning infinite sets.
Take S to be the set of real numbers from 0 to 1000.
Presumably there is a 1 to 1 function mapping S onto the set of real numbers from 0 to 10, and I expect f(x) = x/100 to be one such function.
But suppose the function was f(x) = x/n, where n is arbitrarily large?
Surely this means that the infinite set S can be mapped on to any real interval, even of infinitesimal size?
In the limit as n à ∞, would that mean that S can be mapped onto any single real number? But then it would no longer be a 1 to 1 function, but ∞ à 1?
Going the other way, the set S could have been mapped onto ever ‘wider’ subsets of the Reals, 0 to 1,000,000,000, then 0 to a googolplex, eventually 0 à ∞, or -∞,à ∞?
Ultimately, couldn’t the entire infinite set of Reals be mapped onto an infinitesimal interval at any single chosen real number? What would happen in the limit as the interval tends to 0? Would it be as though any real number somehow ‘contains’ all real numbers? As though a point can give rise to infinity, and infinity can collapse into a point? [Would similar logic hold for complex numbers?]
If you can sort out my understanding, perhaps by pointing me in the direction of a few articles for me to read, I would be very grateful.
Thank you very much for your time.