Incompleteness theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 15/01/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems


00:01:41 1 Formal systems: completeness, consistency, and effective axiomatization
00:03:27 1.1 Effective axiomatization
00:04:40 1.2 Completeness
00:07:41 1.3 Consistency
00:09:05 1.4 Systems which contain arithmetic
00:11:28 1.5 Conflicting goals
00:13:43 2 First incompleteness theorem
00:16:16 2.1 Syntactic form of the Gödel sentence
00:19:04 2.2 Truth of the Gödel sentence
00:21:57 2.3 Relationship with the liar paradox
00:23:39 2.4 Extensions of Gödel's original result
00:26:32 3 Second incompleteness theorem
00:28:48 3.1 Expressing consistency
00:30:15 3.2 The Hilbert&#8211;Bernays conditions
00:31:43 3.3 Implications for consistency proofs
00:35:04 4 Examples of undecidable statements
00:36:03 4.1 Undecidable statements provable in larger systems
00:39:32 5 Relationship with computability
00:41:10 6 Proof sketch for the first theorem
00:42:53 6.1 Arithmetization of syntax
00:45:29 6.2 Construction of a statement about "provability"
00:47:14 6.3 Diagonalization
00:50:26 6.4 Proof via Berry's paradox
00:53:53 6.5 Computer verified proofs
00:58:36 7 Proof sketch for the second theorem
00:59:44 8 Discussion and implications
01:00:55 8.1 Consequences for logicism and Hilbert's second problem
01:02:19 8.2 Minds and machines
01:02:43 8.3 Paraconsistent logic
01:04:11 8.4 Appeals to the incompleteness theorems in other fields
01:08:37 9 History
01:09:39 9.1 Announcement
01:10:54 9.2 Generalization and acceptance
01:13:22 9.3 Criticisms
01:16:44 9.3.1 Finsler
01:18:33 9.3.2 Zermelo
01:18:42 9.3.3 Wittgenstein
01:19:57 10 See also
01:21:15 11 References
01:25:07 11.1 Articles by Gödel



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SUMMARY
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Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.  These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers.  For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.