#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins

Joel David Hamkins, a mathematician and philosopher, discusses the profound implications of Cantor's discovery of different sizes of infinity, which sparked a theological crisis and backlash from contemporaries like Kronecker. The emergence of paradoxes, such as Russell's paradox, further threatened mathematical consistency. Hamkins traces the historical context of infinity from Aristotle's potentialist view to Galileo's paradox, which illustrated the confusion surrounding infinite sets.

The conversation explores equinumerosity, demonstrating how different lengths can have a one-to-one correspondence, as seen in Hilbert's Hotel, where an occupied hotel with infinitely many rooms can still accommodate new guests. This principle is exemplified by the ability to combine countably infinite sets without altering their countable nature. The discussion also highlights prime factorization's uniqueness and its application in assigning odd numbers to train passengers.

Hamkins visualizes countable infinity through the integer lattice and discusses the transition to uncountable infinities, emphasizing Cantor's proof that the set of real numbers is uncountable. He introduces algebraic and transcendental numbers, noting that most real numbers are transcendental, with famous examples like π. A playful proof about "interesting" numbers leads to a contradiction regarding the existence of a "boring" number, reinforcing Cantor's diagonal argument that shows the infinity of real numbers is larger than that of natural numbers.

The conversation delves into Gödel's incompleteness theorems, which challenge Hilbert's program aimed at establishing a reliable foundation for mathematics. Gödel's first theorem asserts that any theory containing a certain level of arithmetic will always be incomplete, while the second emphasizes that such a theory cannot demonstrate its own consistency. This leads to skepticism about the trustworthiness of theories claiming self-consistency.

Hamkins discusses the significance of the axiom of choice within Zermelo-Fraenkel set theory (ZFC), which serves as the foundation for modern mathematics. The conversation touches on Russell's paradox and Cantor's proof of uncountability, emphasizing that the power set of any set is strictly larger than the original set. The implications of Cantor's work led to a restructuring of mathematics, establishing set theory as a foundational framework.

The continuum hypothesis, a central question for Hilbert, reflects his quest for a comprehensive mathematical framework. Cantor's contributions are emphasized, particularly his proofs regarding open and closed sets. The discussion highlights the independence of the continuum hypothesis from ZFC axioms, with Gödel and Cohen demonstrating its consistency. The concept of a "set theoretic multiverse" emerges, suggesting multiple mathematical universes exist rather than a single true mathematics.

Hamkins also explores surreal numbers, introduced by John Conway, which unify various number systems and form a comprehensive mathematical system. The conversation touches on cellular automata and the unpredictability of complex systems, paralleling the halting problem. The discussion on complexity theory highlights NP complete problems and the ongoing curiosity surrounding the P vs NP question.

The speaker reflects on the nature of mathematical progress, emphasizing the importance of collaboration and the emotional challenges of solitary work. They express skepticism about AI's current effectiveness in mathematical reasoning, highlighting the need for structured input for better collaboration with AI systems. The conversation concludes with a philosophical exploration of the distinction between truth and proof, illustrating the complexity of understanding knowledge and reality.