Cantor's Diagonal Argument and the Axiom of Choice: The Paradoxes of Infinite Sets

Overview

In this exploration by Veritasium, the seemingly simple rules of mathematics reveal counterintuitive truths about infinity. The video traces how the real numbers defy countable ordering, introduces Cantor's diagonal argument, and explains the birth of the axiom of choice through Zermelo's work. It also surveys famous paradoxes like the Vitali set and the Banach–Tarski style constructions that challenge our intuition about size and measurability. The discussion culminates in Gödel and Cohen showing that the axiom of choice cannot be proven or disproven from other axioms, leaving math in a landscape where choosing to assume the axiom has profound consequences for how we reason about infinite sets.

Introduction: Why Infinity Feels Intuitive Yet Strangely False

Veritasium introduces a core mathematical tension: choosing from an infinite set is not as straightforward as picking the smallest element. The real numbers stretch to negative infinity, making a natural order elusive and setting the stage for deep questions about how infinity can be compared and organized.

Cantor's Diagonal Argument and the Birth of Uncountable Infinities

Georg Cantor showed there are infinities that cannot be put into a one-to-one correspondence with the natural numbers. By assuming a complete list of all real numbers in [0,1] and constructing a new number that differs from every listed number in a diagonal digit, Cantor produced a real number not on the list. This diagonalization proves there are more real numbers than natural numbers, introducing the distinction between countable and uncountable infinities and reshaping our understanding of infinity as a hierarchy, not a single size.

Well-Ordering and the Axiom of Choice

Cantor aspired to well order all sets, but finding a starting point for uncountable sets proved impossible with the existing framework. This led to Zermelo formulating the Axiom of Choice, which guarantees the ability to select one element from each of an infinite collection of nonempty sets, even without a concrete rule. Using the Axiom of Choice, Zermelo established that every set could be well ordered, linking Cantor's ideas to a formal proof system.

Controversies, Paradoxes, and the Foundations Crisis

The Axiom of Choice produced unsettling results, including non-measurable sets like the Vitali set, which resist traditional notions of size and probability. Infamous constructions, such as the Banach–Tarski paradox in spirit, demonstrate how infinite processes can yield counterintuitive outcomes. Philosophers and mathematicians debated whether the axiom should be assumed, since it enables powerful but nonconstructive proofs that do not provide explicit choices.

Godel and Cohen: The Modern Landscape

Kurt Godel showed that the Axiom of Choice is consistent relative to the other axioms, while Paul Cohen demonstrated that its negation is also consistent, depending on the chosen axiomatic system. This established a dual universe: mathematics remains coherent with or without the axiom, contingent on which system one adopts. Veritasium emphasizes that the decision to include the axiom is a convention that guides mathematical reasoning and problem solving.

Conclusion

Despite the counterintuitive consequences, the Axiom of Choice remains a central, practical tool in modern mathematics. It simplifies reasoning about infinite structures and enables concise proofs, even as it invites ongoing exploration of what is provable when choice is assumed or avoided. The video closes by reflecting on the broader philosophical and practical implications for how we understand and use infinity in mathematics.