Interview
Infinity, Paradoxes, Gödel Incompleteness & the Mathematical Multiverse | Lex Fridman Podcast #488
- Expects to undertake challenging travel in the coming year and plans to dedicate upcoming work to students with a focus on elegant proofs revealing mathematical beauty.
- Writes a continuing series of essays on the Surreal Numbers via his Substack platform, Infinitely More, with the ambition to establish the system as a fundamental framework for mathematics and science, a status it has not yet attained despite John Conway's hopes.
- Predicts that mathematical practices a thousand years hence will be completely unrecognizable to current practitioners, while anticipating that the field will experience huge progress and improved understanding over that millennium.
- Anticipates that physics may eventually provide a convincing account of physical existence, resolving current profound mysteries, while the distinction between truth and proof will reveal fascinating mysteries across physics, psychology, sociology, politics, and geopolitics.
- Projects that computer procedures could correctly decide almost every instance of the halting problem due to the asymptotic density of programs failing for trivial reasons, and expects feasible algorithms to solve almost every instance of NP-complete problems.
- Maintains skepticism regarding current AI systems for mathematical reasoning, predicting they will frequently produce incorrect answers by simulating logical correctness without achieving genuine mathematical insight or actual proof.
- Expects the game of life to contain computationally undecidable questions, specifically equating the question of cell activation to the halting problem.
- Foresees that set theory will remain divided between universe view (monism), which seeks the nature of the one true set theoretic universe exemplified by Hugh Woodin's ultimate L, and pluralist view, which directs research toward interactions between different universes and forcing extensions.
- Anticipates that set-theoretic geology, originating from a pluralist perspective, will be adopted by universe view researchers to aid in discovering the nature of the one true universe.
- Predicts that the Continuum Hypothesis can be toggled like a light switch depending on the set theoretic universe inhabited, characterizing the answer as a pluralist choice rather than a singular truth.
- Notes that infinite chess positions can possess game values corresponding to every countable ordinal, a result that has improved dramatically since its initial discovery.
- Expects transfinite ordinals to continue serving as the foundation for transfinite recursive constructions, including the V hierarchy and Gödel's constructible universe.
- Suggests that if the axiom of choice were inconsistent, that inconsistency would already exist independently of the axiom, implying the axiom is not the source of any such inconsistency.
- Maintains a personal intent to live entirely within the platonic realm regarding reality, while expressing acceptance of the mixture of hate and love received from his audience.
- Notes that an ancient mathematician like Archimedes might be able to understand some current mathematical developments, contrasting with the anticipated incomprehensibility of future mathematics.