Part 5/9:
Every formal mathematical system begins with a predetermined set of axioms. The choice of axioms is subjective, revealing that the structure of mathematics is not entirely objective as it is typically perceived. This contradiction surfaces in discussions about how various mathematical frameworks can yield consistent yet non-contradictory conclusions from different foundational principles.
Gödel's first Incompleteness Theorem asserts that in any sufficiently complex formal system—one that encompasses basic arithmetic—there exist true statements that cannot be proven from the axioms of that system. This assertion fundamentally overturns the assumption that one could eventually derive all mathematical truths systematically and completely.