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Mathematical logic

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Улугбек{{for|Quine's theory sometimes called "Mathematical Logic"|New Foundations}}
{{redirect|Mathematical formalism|the philosophical view|Formalism (philosophy of mathematics)}}
'''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]].<ref>Undergraduate texts include Boolos, Burgess, and Jeffrey [[#CITEREFBoolosBurgessJeffrey2002|(2002)]], [[Herbert Enderton|Enderton]] [[#CITEREFEnderton2001|(2001)]], and Mendelson [[#CITEREFMendelson1997|(1997)]]. A classic graduate text by Shoenfield [[#CITEREFShoenfield2001|(2001)]] first appeared in 1967.</ref> The unifying themes in mathematical logic include the study of the expressive power of [[formal system]]s and the [[Deductive reasoning|deductive]] power of formal [[Mathematical proof|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, particularупlyparticularly [[first-order logic]], and [[definable set|definability]]. In computer science (particularly in the [[ACM Computing Classification System|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 [[axiom]]atic frameworks for [[geometry]], [[arithmetic]], and [[analysis]]. In the early 20th century it was shaped by [[David Hilbert]]'s [[Hilbert's program|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.