A sumptuous banquet of mathematical tools. Not that I'm a real mathematician but I think I can add one more. Roughly, Suppose you have a group of axioms and y

A sumptuous banquet of mathematical tools. Not that I'm a real mathematician but I think I can add one more. Roughly, Suppose you have a group of axioms and you want to show that a minimal set/system satisfying those axioms exists. Define your objects as the object that exist according to the axioms plus every application of an operation specified by the axioms. Then show the set itself is closed under those operations. The sort of approach used to define the real numbers, for Godel's Constructible Universe [1], the Löwenheim–Skolem theorem [2] and a variety of other places. [1] https://en.wikipedia.org/wiki/Constructible_universe https://en.wikipedia.org/wiki/Constructible_universe [2] https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...