The fun part is that what we invented is a set of rules that governs what else we may or may not invent! That is, once you agree on what logic is and what some

The fun part is that what we invented is a set of rules that governs what else we may or may not invent! That is, once you agree on what logic is and what some axioms are, you can show that certain statements can't be added as axioms. Similarly, once you've decide on your axioms, and your set of logical tools, you will find some things (like irrational numbers), that you wouldn't have introduced directly. It's pretty cool!