Mathematical logic

formal logicsymbolic logiclogiclogicianformalmathematical logicianlogicalmathematical formalismsymboliclogicians
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.wikipedia
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Set theory

axiomatic set theoryset-theoreticset
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

Foundations of mathematics

foundation of mathematicsfoundationsfoundational
It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.
The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science.

Proof theory

proof-theoreticProof-theoreticallyderive
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

Logic

logicianlogicallogics
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.

Model theory

modelmodelsmodel-theoretic
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

Theoretical computer science

theoretical computer scientisttheoreticalcomputer science
It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.

Definable set

definabledefinabilitydefining
These areas share basic results on logic, particularly first-order logic, and definability.
In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the first-order language of that structure.

Reverse mathematics

arithmetical transfinite recursionbounded reverse mathematicsconstructive reverse mathematics
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.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.

David Hilbert

HilbertHilbert, DavidD. Hilbert
In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories.
Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Metamathematics

metamathematicalmeta-mathematicalmeta-mathematics
It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.
Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap.

Boolean algebra

Booleanboolean logiclogic
"Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last [nineteenth] century with the aid of an artificial notation and a rigorously deductive method."
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

Löb's theorem

Löb's rule
Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic.
In mathematical logic, Löb's theorem states that in any formal system F with Peano arithmetic (PA), for any formula P, if it is provable in F that "if P is provable in F then P is true", then P is provable in F.

Categorical logic

internal languagecategorical semanticsCategorical
The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic.
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.

Logic in China

ChinaChinese logical
Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world.
Formal logic in China has a special place in the history of logic due to its repression and abandonment—in contrast to the strong ancient adoption and continued development of the study of logic in Europe, India, and the Islamic world.

Gottlob Frege

FregeFrege, GottlobFregean
Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic.
Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.

Bertrand Russell

RussellRussell, Bertrand16 Questions on the Assassination
Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century.
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

Topos

topos theorytopoielementary topos
These foundations use toposes, which resemble generalized models of set theory that may employ classical or nonclassical logic.
The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

Forcing (mathematics)

forcinggeneric extensionCohen
The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.
Forcing has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory.

History of logic

logicmodern logicGreece
Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world.
Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.

Giuseppe Peano

PeanoPeano, Giuseppe
Giuseppe Peano (1889) published a set of axioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Boole and Schröder but adding quantifiers.
The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation.

Mathematics

mathematicalmathmathematician
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic.

Mathematical induction

inductioninductivelycomplete induction
Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.

Ernst Schröder

SchröderSchroederSchröder, Ernst
From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes.
He is a major figure in the history of mathematical logic (a term he may have invented), by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce.

Kurt Gödel

GödelGödel, Kurt Gödel, Kurt
Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency.
Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic.

Hilbert's program

programattemptDavid Hilbert
In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories.
Many current lines of research in mathematical logic, such as proof theory and reverse mathematics, can be viewed as natural continuations of Hilbert's original program.