The ZFC axiom set [1] doesn't define a made-up world. Actually, once you understand the axioms (they're somewhat difficult in notation and definitions, but the

The ZFC axiom set [1] doesn't define a made-up world. Actually, once you understand the axioms (they're somewhat difficult in notation and definitions, but the ideas behind are understandable) they make sense. There are some axioms made to avoid paradoxes (notably ZFC3 [2]), but the majority describe how we understand the world. It'd be difficult to imagine a world where two sets aren't equal even when they have the same elements, or where you couldn't construct a set that contains all elements of two other sets. How's this different to a set of facts as you say, like one unit + one unit is two units? It's not. I'm not sure of this, but I'd say you could construct a set of axioms based on those facts that is compatible with the ZFC axiom set. So, why choose ZFC? Because it's simple to define (how'd you formally define what is a unit, and what does the + operation mean?) and because it's more or less consistent with our world. We can extrapolate the things we find while doing mathematics to the real world. And, sometimes, we can discover things of the real world based on findings on things that don't exist. Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments [3]. We approach all of mathematics like this because its easier to study everything within the same system. And it's important that we keep a consistent underlying system so we can be sure that everything we discover based on the set of axioms and rules we've defined is true in a mathematical sense; and then those discoveries could describe our world. Wouldn't we have a consistent system, we could end up with contradictions. If we extracted mathematical concepts from the real world, we wouldn't be sure if a discovery we make from those concepts can be applied to the real world. TL;DR: Our set of axioms describes properly the real world, is simple and consistent enough and that enables us to infer new ideas that can be applicable to the real world too. [1] http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_th... [2] http://en.wikipedia.org/wiki/Axiom_schema_of_specification http://en.wikipedia.org/wiki/Axiom_schema_of_specification [3] http://en.wikipedia.org/wiki/Complex_number#Electromagnetism_and_electrical_engineering http://en.wikipedia.org/wiki/Complex_number#Electromagnetism...