= Math

well-ordering:: every subset has a minimum
partial ordering:: ⊆
total ordering:: every pair of elements allow comparison

It was proven that
  … the ZF axioms do not imply the axiom of choice
  … the ZF axioms do not contradict the axiom of choice

Once you can construct the integers, you run into Godel's incompleteness theorem and have contradictions or incompleteness

ZFC and the definition of addition ⇒ 1+1=2

Peano axioms are weaker than ZFC (e.g. no overcountable elements) and are not in contradiction to ZFC

Dedekind cuts: e.g.
  sqrt(2) := {s ∈ ℚ | s² < 2 ∨ s < 0}

How to construct ℝ from ℚ?
  approach 1:: Dedekind cuts with axiom of exclusion
  approach 2:: Cauchy series (remember:  ∃ limit, then it is a Cauchy series; Cauchy series only converge in some spaces)