>>10282927Constructing the rationals is extremely easy. You define a field, and then you write down "the smallest field with char 0".
Constructing the reals is, on the other hand, a bit of a hassle. So there are a couple approaches:
You take the set of Cauchy-sequences of the rational numbers, and take the equivalence class of functions with the same limit. Sum and multiplication are defined pointwise. So the number one looks like the sequence {1, 1, 1, 1, .....}, pi looks like {3, 3.1, 3.14, ....}, and so on.
A Dedekind cut is a set that, if it includes some rational x, also includes every rational smaller than x. We set every Dedekind cut to be a real number, and sum and multiplication are set trickery.
The axiomatic approach is basically taking a field, and saying that it satisfies the Dedekind axiom.
Ultimately, those you asked are true because they're true in the rationals.
