Select a foundational axiom system in mathematics (such as Zermelo-Fraenkel set theory, Peano axioms, Euclid's axioms, etc.). Explain the historical context in which this axiom system was developed. Detail each axiom in the system and explain its purpose. Discuss the implications of these axioms for mathematics as a whole. Analyze any known limitations, controversies, or alternatives to this axiom system. Finally, explore how this axiom system continues to influence mathematical thinking and research today.