There Is a Number That Answers Every Question and You Are Forbidden to Read It

🔴 Support the channel and get exclusive content:   / meridianlabs   In two thousand one, three researchers at the University of Auckland certified the first sixty four binary digits of a number that cannot be computed. Chaitin's constant, Omega, is the probability that a randomly assembled computer program eventually halts. Its digits encode the answer to Goldbach's conjecture, the Riemann hypothesis, and whether the axioms underneath modern mathematics contain a contradiction. A few thousand of those bits would settle problems that have stood for centuries. And Gregory Chaitin proved in nineteen seventy five that no consistent formal system can ever determine more than a fixed, finite number of them. The allowance is set in advance by how complicated your axioms are. Robert Solovay later built a version for which set theory cannot determine a single bit. This is the story of a number that is perfectly well defined, provably real, and permanently unreadable, and of the question underneath it: whether most mathematical truths are true for any reason at all. Featuring Alan Turing's nineteen thirty six paper, Kurt Godel's incompleteness theorems, Emil Post, Hilbert's tenth problem, the busy beaver function, the five hundred forty nine state machine whose halting is independent of ZFC, and the closing return to Mills' constant. 0:00 Intro 2:00 The Number That Answers Everything 4:25 The Allowance 11:05 Does It Stop 17:15 A Phrase Turing Never Wrote 19:45 The Man Who Interrupted Himself 26:40 The Quarantine Fails 33:10 Three Machines 40:30 The Last Busy Beaver 46:55 A Teenager At IBM 50:50 The Probability That It Halts 59:00 Knowledge Without Proof 1:05:40 The Librarian's Paradox 1:12:35 Sixty Five Pounds 1:20:55 Sixty Four Bits In Auckland 1:27:15 Not A Single Bit 1:35:30 The Ground Floor 1:44:30 True For No Reason 1:54:35 The Mirror And The Book Grounded in primary sources, including: Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem (1936) — Proceedings of the London Mathematical Society Alonzo Church, An Unsolvable Problem of Elementary Number Theory (1936) — American Journal of Mathematics Kurt Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I (1931) — Monatshefte für Mathematik und Physik Martin Davis, Computability and Unsolvability (1958) — McGraw-Hill Martin Davis, Hilary Putnam and Julia Robinson, The Decision Problem for Exponential Diophantine Equations (1961) — Annals of Mathematics Tibor Radó, On Non-Computable Functions (1962) — Bell System Technical Journal Yuri Matiyasevich, Enumerable Sets Are Diophantine (1970) — Soviet Mathematics Doklady Gregory Chaitin, A Theory of Program Size Formally Identical to Information Theory (1975) — Journal of the ACM Gregory Chaitin, Algorithmic Information Theory (1987) — Cambridge University Press Panu Raatikainen, On Interpreting Chaitin's Incompleteness Theorem (1998) — Journal of Philosophical Logic Robert M. Solovay, A Version of Omega for Which ZFC Cannot Predict a Single Bit (2000) — Springer Antonín Kučera and Theodore A. Slaman, Randomness and Recursive Enumerability (2001) — SIAM Journal on Computing Cristian S. Calude, Michael J. Dinneen and Chi-Kou Shu, Computing a Glimpse of Randomness (2002) — Experimental Mathematics Adam Yedidia and Scott Aaronson, A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory (2016) — Complex Systems W. H. Mills, A Prime-Representing Function (1947) — Bulletin of the American Mathematical Society #Chaitin #HaltingProblem #Incompleteness #Mathematics #MathDocumentary