The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy
Explore Gödel’s Incompleteness Theorem, a discovery which changed what we know about mathematical proofs and statements. -- Consider the following sentence: “This statement is false.” Is that true? If so, that would make the statement false. But if it’s false, then the statement is true. This sentence creates an unsolvable paradox; if it’s not true and it’s not false– what is it? This question led a logician to a discovery that would change mathematics forever. Marcus du Sautoy digs into Gödel’s Incompleteness Theorem. Lesson by Marcus du Sautoy, directed by BASA. Support Our Non-Profit Mission ---------------------------------------------- Support us on Patreon: http://bit.ly/TEDEdPatreon Check out our merch: http://bit.ly/TEDEDShop ---------------------------------------------- Connect With Us ---------------------------------------------- Sign up for our newsletter: http://bit.ly/TEDEdNewsletter Follow us on Facebook: http://bit.ly/TEDEdFacebook Find us on Twitter: http://bit.ly/TEDEdTwitter Peep us on Instagram: http://bit.ly/TEDEdInstagram ---------------------------------------------- Keep Learning ---------------------------------------------- View full lesson: https://ed.ted.com/lessons/the-paradox-at-the-heart-of-mathematics-godel-s-incompleteness-theorem-marcus-du-sautoy Dig deeper with additional resources: https://ed.ted.com/lessons/the-paradox-at-the-heart-of-mathematics-godel-s-incompleteness-theorem-marcus-du-sautoy#digdeeper Animator's website: https://basaestudio.com ---------------------------------------------- Thank you so much to our patrons for your support! Without you this video would not be possible! Dwight Schrute, Dianne Palomar, Marin Kovachev, Fahad Nasser Chowdhury, Penelope Misquitta, Hans Peng, Gaurav Mathur, Erik Biemans, Tony, Michelle, Katie and Josh Pedretti, Sunny Patel, Hoai Nam Tran, Stina Boberg, Kack-Kyun Kim, Michael Braun-Boghos, Ken, zjweele13, Jurjen Geleijn, Anna-Pitschna Kunz, Edla Paniguel, Elena Crescia, Thomas Mungavan, Jaron Blackburn, Venkat Venkatakrishnan, ReuniteKorea, Aaron Henson, Rohan Gupta, Begum Tutuncu, Ever Granada, Mikhail Shkirev, Brian Richards, Cindy O., Jørgen Østerpart, Tyron Jung, Carolyn Corwin, Carsten Tobehn, Katie Dean, Ezgi Yersu, Gerald Onyango, alessandra tasso, Côme Vincent, Doreen Reynolds-Consolati, Manognya Chakrapani, Ayala Ron, Samantha Chow, Eunsun Kim, Phyllis Dubrow, Ophelia Gibson Best, Paul Schneider, Joichiro Yamada and Henrique 'Sorín' Cassús.
The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy
Gödel's Incompleteness Theorem
This video discusses the limitations of mathematical proofs and how Kurt Gödel's discovery of self-referential statements led to the development of his Incompleteness Theorem, which introduced a new class of mathematical statement that cannot be proven or disproven within a given set of axioms.
Gödel's Discovery
- Kurt Gödel discovered the limitations of mathematical proofs in the early 20th century.
- A proof is a logical argument that demonstrates why a statement about numbers is true.
- Proofs are built on axioms, which are undeniable statements about the numbers involved.
- Every system built on mathematics, from basic arithmetic to complex proofs, is constructed from axioms.
Self-Referential Statements
- Mathematicians used axiomatic systems to prove or disprove mathematical claims with total certainty since ancient Greece.
- Through coding, mathematics can talk about itself.
- Gödel created the first self-referential mathematical statement by writing "This statement cannot be proved" as an equation.
- If this statement were false, it would have a proof. But if it has a proof, then it must be true. Therefore, it cannot be false and must be true.
Incompleteness Theorem
- Gödel's Incompleteness Theorem introduces an entirely new class of mathematical statement where true statements can either be provable or unprovable within a given set of axioms.
- Unprovable true statements exist in every axiomatic system making it impossible to create a perfectly complete system using mathematics.
- Adding unprovable statements as new axioms to an enlarged mathematical system introduces new unprovably true statements.
- Gödel's theorem opened as many doors as it closed, and knowledge of unprovably true statements is now a fundamental part of mathematics.
The Impact of Gödel's Incompleteness Theorem
This section discusses how Gödel's incompleteness theorem impacted the field of mathematics.
Gödel's Incompleteness Theorem
- Some mathematicians dedicate their careers to identifying provably unprovable statements.
- Mathematicians may have lost some certainty, but they can embrace the unknown at the heart of any quest for truth.