Writing
3 posts tagged "Foundations"
We explore the Axiom of Choice, its equivalence with Zorn's Lemma and the Well-Ordering Principle, the controversies surrounding it, and its indispensable role throughout modern mathematics.
We explain Gödel's two incompleteness theorems — why any consistent formal system strong enough to express arithmetic contains true statements it cannot prove, and why it cannot prove its own consistency.
We present Cantor's diagonal argument proving that the real numbers are uncountable, explore its consequences for set theory, and examine how it forever changed our understanding of infinity.