Enjoy discussing, helps me learn too.

Another way to phrase it, is you have a short set of rules: linear order, upper bounds, dense countable subset..., that you hone in on as you begin to formalize what you imagine reals to be, and then there are two interesting questions. Are these rules consistent (is there at least one mathematical model satisfying them, or do they somehow lead to contradictions), and are they narrow they don't allow for other interpretations. Ie, have you nailed down a set of rules such that you can declare they unambiguously define THE real numbers.

For instance, if you drop the countable dense subset requirement, you get nonstandard models that also satisfy the rest of the rules.

I'm blathering...