The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy
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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.
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Prior to Gödel’s mathematical discoveries, mathematicians had believed the foundation of mathematical systems to be both complete and consistent. But the challenge was to prove that this was true. Known as Hilbert's program, these two features were intended to solve inconsistencies and paradoxes in the foundation of math. The first feature, completeness, is based on the assumption that true mathematical statements should be provable. The second feature, consistency, holds that if a statement is proved true, its opposite cannot also be true. Mathematician David Hilbert simplified the consistency and completeness of all mathematics to questions of basic arithmetic. His hope was that this simplification would allow mathematics to prove itself complete and consistent. Gödel’s discovery unraveled Hilbert’s hope. His discovery highlights the difference between what is true and what is provable.
Gödel’s incompleteness theorems state that within any system for arithmetic there are true mathematical statements that can never be proved true. The first step was to code mathematical statements into unique numbers known as Gödel’s numbers; he set 12 elementary symbols to serve as vocabulary for expressing a set of basic axioms. These symbols assigned specific numbers to unique statements and formulas. By doing so, Gödel was able to make self-referential statements using numbers, allowing him to write “this statement is unprovable”
While this might seem as a paradox, Gödel proved that it’s a true statement. If the statement is considered false, then that would mean the statement is provable, which would mean it’s true. To learn more about Gödel’s incompleteness theorem, check out these links by the Marcus du Sautoy, the educator of this lesson:- https://www.youtube.com/watch?v=lLsm-ceqyio- https://www.youtube.com/watch?v=O4ndIDcDSGc
To find out more about the power of mathematical proof as one of humanity's greatest shortcuts to the truth check out Thinking Better: the Art of the Shortcut. In this book du Sautoy outlines some of the powerful logical arguments and proofs that mathematicians have gathered over the two thousand years we have been doing mathematics in order to get us to our destination in the fastest and most efficient manner.https://harpercollins.co.uk/products/thinking-better-the-art-of-the-shortcut-marcus-du-sautoy?variant=32750330118222https://www.basicbooks.com/titles/marcus-du-sautoy/thinking-better/9781541600362/
Gödel’s incompleteness theorems state that within any system for arithmetic there are true mathematical statements that can never be proved true. The first step was to code mathematical statements into unique numbers known as Gödel’s numbers; he set 12 elementary symbols to serve as vocabulary for expressing a set of basic axioms. These symbols assigned specific numbers to unique statements and formulas. By doing so, Gödel was able to make self-referential statements using numbers, allowing him to write “this statement is unprovable”
While this might seem as a paradox, Gödel proved that it’s a true statement. If the statement is considered false, then that would mean the statement is provable, which would mean it’s true. To learn more about Gödel’s incompleteness theorem, check out these links by the Marcus du Sautoy, the educator of this lesson:- https://www.youtube.com/watch?v=lLsm-ceqyio- https://www.youtube.com/watch?v=O4ndIDcDSGc
To find out more about the power of mathematical proof as one of humanity's greatest shortcuts to the truth check out Thinking Better: the Art of the Shortcut. In this book du Sautoy outlines some of the powerful logical arguments and proofs that mathematicians have gathered over the two thousand years we have been doing mathematics in order to get us to our destination in the fastest and most efficient manner.https://harpercollins.co.uk/products/thinking-better-the-art-of-the-shortcut-marcus-du-sautoy?variant=32750330118222https://www.basicbooks.com/titles/marcus-du-sautoy/thinking-better/9781541600362/
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