>Definition 2.1.1 A line is a 1D Hausdorff space extending infinitely far in
both directions.
What do you mean by a 1D space? The usual meaning is a topological or smooth manifold, but these notions depend on the real numbers already being constructed.
>Definition 2.1.3 The real number line is a number line given the label “real.”
So if I take the set of all rational numbers and put them on a line, will that constitute a real number line?
>Definition 2.1.4 If x is a cut in a line, then (??, ?) = (??, x] ? (x, ?) .
With this definition it looks like cuts correspond bijectively with points on the line
>Definition 2.1.5 A real number x ? R is a cut in the real number line.
This doesn't define anything. As I said before, the line could consist only of rational numbers and then the cuts by your definition would also correspond to rational numbers, hence be countable, contradicting a well-known result of Cantor and demonstrating that your definition is wrong.
>Axiom 2.1.6 Real numbers are such that
>?x, y ? R s.t. x 6= y ?n ? N s.t.
Couldn't copy it right, but typically you would prove this. Also the rational numbers satisfy the same statement. But props to you for admitting 0.999...=1
>Axiom 2.1.7 Real numbers are represented in algebraic interval notation as R = (??,?) . In other words, x ? R if x is both less than infinity and greater than minus
infinity. The connectedness of R is explicit in the interval notation.
>The connectedness of R is explicit in the interval notation.
No it doesn't. Connectedness is a topological property, and the rationals are disconnected in the usual order topology, yet they also can be described as (??,?) .
Introducing notation does not constitute a proof of anything. Notation is just there to make it easier to represent concepts or objects that are actually there.
cont.
