Axioms are the foundational assumptions from which formal systems of mathematics are built. Some systems of axioms are unable to prove the truth or falsity of some statements within that system. But you can add such statements to your set of axioms to form a new, larger formal system, which in turn has other indeterminate statements, and so on, thus building, in GP’s terms, an infinite playground of mathematics.

Book recommendation: Gödel, Escher, Bach by Douglas Hofstadter.