Mathematical formalism | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-01-15

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Mathematical_logic


00:01:52 1 Subfields and scope
00:03:28 2 History
00:04:28 2.1 Early history
00:04:59 2.2 19th century
00:06:29 2.2.1 Foundational theories
00:10:30 2.3 20th century
00:11:35 2.3.1 Set theory and paradoxes
00:14:08 2.3.2 Symbolic logic
00:16:32 2.3.3 Beginnings of the other branches
00:18:29 3 Formal logical systems
00:19:17 3.1 First-order logic
00:22:37 3.2 Other classical logics
00:25:30 3.3 Nonclassical and modal logic
00:26:41 3.4 Algebraic logic
00:27:19 4 Set theory
00:31:13 5 Model theory
00:33:10 6 Recursion theory
00:34:43 6.1 Algorithmically unsolvable problems
00:36:15 7 Proof theory and constructive mathematics
00:37:58 8 Applications
00:39:28 9 Connections with computer science
00:40:56 10 Foundations of mathematics
00:45:29 11 See also



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SUMMARY
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Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.  It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.  The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.  These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.
Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.