Summary of Gödel's Incompleteness Theorem - Numberphile

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Gödel's incompleteness theorem says that there will always be some mathematical statements that we can't prove. This means that there's always going to be a gap between truth and proof, and some truths may be hidden from us. Even if we expand our mathematics, there will always be something we're missing.

• 00:00:00 Gödel's incompleteness theorem states that there may be true statements about numbers that cannot be proven within any given mathematical system. This implies that there is a gap between truth and proof, and raises the possibility that some mathematical truths may be hidden from us.
• 00:05:00 Gödel's Incompleteness Theorem states that there is a statement in mathematics that cannot be proved from the axioms of that system. However, by assuming that mathematics is consistent, it is possible to create a statement that is true but cannot be proven. This statement is called a Gödel statement and, as long as the system remains consistent, it can be added to the axioms as an axiom.
• 00:10:00 The Gödel incompleteness theorem states that certain mathematical statements are unprovable, and that no matter how much we expand our mathematics, we will always be missing something. This raises the question of whether certain mathematical statements are really just inaccessible to us, or if there might be a way to prove them even if we don't know all of the axioms.