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Mathematical logic

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

For Quine's theory sometimes called "Mathematical Logic", see New Foundations. For other uses, see Logic (disambiguation).

Since its inception, mathematical logic has both contributed to and 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.

Applications[edit]

"Mathematical logic has been successfully applied not only to mathematics and its foundations (G. Frege, B. Russell, D. Hilbert, P. Bernays, H. Scholz, R. Carnap, S. Lesniewski, T. Skolem), but also to physics (R. Carnap, A. Dittrich, B. Russell, C. E. Shannon, A. N. Whitehead, H. Reichenbach, P. Fevrier), to biology (J. H. Woodger, A. Tarski), to psychology (F. B. Fitch, C. G. Hempel), to law and morals (K. Menger, U. Klug, P. Oppenheim), to economics (J. Neumann, O. Morgenstern), to practical questions (E. C. Berkeley, E. Stamm), and even to metaphysics (J. [Jan] Salamucha, H. Scholz, J. M. Bochenski). Its applications to the history of logic have proven extremely fruitful (J. Lukasiewicz, H. Scholz, B. Mates, A. Becker, E. Moody, J. Salamucha, K. Duerr, Z. Jordan, P. Boehner, J. M. Bochenski, S. [Stanislaw] T. Schayer, D. Ingalls)."[47] "Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)."[47]

Argument

Informal logic

Universal logic

Knowledge representation and reasoning

Logic

List of computability and complexity topics

List of first-order theories

List of logic symbols

List of mathematical logic topics

List of set theory topics

Mereology

Propositional calculus

Well-formed formula

Walicki, Michał (2011). Introduction to Mathematical Logic. : World Scientific Publishing. ISBN 9789814343879.

Singapore

; Burgess, John; Jeffrey, Richard (2002). Computability and Logic (4th ed.). Cambridge University Press. ISBN 9780521007580.

Boolos, George

Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London, Oxford, New York City: . ISBN 9780198880875. Zbl 0251.02001.

Oxford University Press

Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). MA: Academic Press. ISBN 978-0-12-238452-3.

Boston

Fisher, Alec (1982). Formal Number Theory and Computability: A Workbook. (suitable as a first course for independent study) (1st ed.). Oxford University Press.  978-0-19-853188-3.

ISBN

Hamilton, A.G. (1988). Logic for Mathematicians (2nd ed.). Cambridge University Press.  978-0-521-36865-0.

ISBN

Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). (2nd ed.). New York City: Springer. ISBN 9780387942582.

Mathematical Logic

Katz, Robert (1964). Axiomatic Analysis. MA: D. C. Heath and Company.

Boston

(1997). Introduction to Mathematical Logic (4th ed.). London: Chapman & Hall. ISBN 978-0-412-80830-2.

Mendelson, Elliott

(2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York City: Springer. doi:10.1007/978-1-4419-1221-3. ISBN 9781441912206.

Rautenberg, Wolfgang

(2003–2004). Mathematical Logic (PDF). Munich: Mathematisches Institut der Universität München. Retrieved 2016-02-24.

Schwichtenberg, Helmut

Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, , 2004, ISBN 0-19-852981-3. Covers logics in close relation with computability theory and complexity theory

Oxford University Press

van Dalen, Dirk (2013). Logic and Structure. Universitext. Berlin: . doi:10.1007/978-1-4471-4558-5. ISBN 978-1-4471-4557-8.

Springer

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Mathematical logic"

Polyvalued logic and Quantity Relation Logic

, a free textbook by P. D. Magnus.

forall x: an introduction to formal logic

, a free textbook by Stefan Bilaniuk.

A Problem Course in Mathematical Logic

Detlovs, Vilnis, and Podnieks, Karlis (University of Latvia), (hyper-textbook).

Introduction to Mathematical Logic.

In the :

Classical Logic by Stewart Shapiro.
First-order Model Theory by Wilfrid Hodges.

Stanford Encyclopedia of Philosophy

In the :

Mathematical Logic
Set Theory & Further Logic
Philosophy of Mathematics

London Philosophy Study Guide

School of Mathematics, University of Manchester, Prof. Jeff Paris’s Mathematical Logic (course material and unpublished papers)