If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then
Why can't the reals, which demands, simply, unlimited precision (this is probably where I'm wrong), be thought of as the infinite set of every possible sum of infinitesimals, where each point along the number line is defined as greater than the next?
If there are an infinite number of rational numbers between 0 and 1, why don't they encompass all the reals, despite being countable?
Why is countability so important? I can't wrap my head around what distinguishes the reals from the rationals, and one infinity from another (as posited by Cantor).