mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. topically, mathematical logic bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.undergraduate texts include boolos, burgess, and jeffrey (2002), enderton (2001), and mendelson (1997). a classic graduate text by shoenfield (2001) first appeared in 1967. 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 gÃ¶del, 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. ==subfields and scope== the handbook of mathematical logic makes a rough division of contemporary mathematical logic into four areas: #set theory #model theory #recursion theory, and #proof theory and constructive mathematics (considered as parts of a single area). each area has a distinct focus, although many techniques and results are shared among multiple areas. the borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. gÃ¶del's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to lÃ¶b's theorem in modal logic. the method
Mathematical_logic