Symbolic Logic

Mathematical logic
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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.[1] 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.

Contents [hide]
1 Subfields and scope
2 History
2.1 Early history
2.2 19th century
2.2.1 Foundational theories
2.3 20th century
2.3.1 Set

References: Walicki, Michał (2011), Introduction to Mathematical Logic, Singapore: World Scientific Publishing, ISBN 978-981-4343-87-9. Boolos, George; Burgess, John; Jeffrey, Richard (2002), Computability and Logic (4th ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-00758-0. Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3. Hamilton, A.G. (1988), Logic for Mathematicians (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-36865-0. Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994), Mathematical Logic (2nd ed.), New York: Springer, ISBN 0-387-94258-0. Katz, Robert (1964), Axiomatic Analysis, Boston, MA: D. C. Heath and Company. Mendelson, Elliott (1997), Introduction to Mathematical Logic (4th ed.), London: Chapman & Hall, ISBN 978-0-412-80830-2. Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6. Schwichtenberg, Helmut (2003–2004), Mathematical Logic, Munich, Germany: Mathematisches Institut der Universität München. Andrews, Peter B. (2002), An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2nd ed.), Boston: Kluwer Academic Publishers, ISBN 978-1-4020-0763-7. Barwise, Jon, ed. (1989), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, North Holland, ISBN 978-0-444-86388-1. Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6. Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7. Shoenfield, Joseph R. (2001) [1967], Mathematical Logic (2nd ed.), A K Peters, ISBN 978-1-56881-135-2. Troelstra, Anne Sjerp; Schwichtenberg, Helmut (2000), Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-77911-1. Katz, Victor J. (1998), A History of Mathematics, Addison–Wesley, ISBN 0-321-01618-1. Morley, Michael (1965), "Categoricity in Power", Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 114, No. 2) 114 (2): 514–538, doi:10.2307/1994188, JSTOR 1994188. Soare, Robert I. (1996), "Computability and recursion", Bulletin of Symbolic Logic (The Bulletin of Symbolic Logic, Vol. 2, No. 3) 2 (3): 284–321, doi:10.2307/420992, JSTOR 420992. Solovay, Robert M. (1976), "Provability Interpretations of Modal Logic", Israel Journal of Mathematics 25 (3–4): 287–304, doi:10.1007/BF02757006. Woodin, W. Hugh (2001), "The Continuum Hypothesis, Part I", Notices of the American Mathematical Society 48 (6). PDF Classical papers, texts, and collections[edit] Burali-Forti, Cesare (1897), A question on transfinite numbers, reprinted in van Heijenoort 1976, pp. 104–111. Dedekind, Richard (1872), Stetigkeit und irrationale Zahlen. English translation of title: "Consistency and irrational numbers". Dedekind, Richard (1888), Was sind und was sollen die Zahlen? Two English translations: 1963 (1901) Hilbert, David (1899), Grundlagen der Geometrie, Leipzig: Teubner, English 1902 edition (The Foundations of Geometry) republished 1980, Open Court, Chicago. Kleene, Stephen Cole (1943), "Recursive Predicates and Quantifiers", American Mathematical Society Transactions (Transactions of the American Mathematical Society, Vol. 53, No. 1) 54 (1): 41–73, doi:10.2307/1990131, JSTOR 1990131. Mancosu, Paolo, ed. (1998), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford: Oxford University Press. Pasch, Moritz (1882), Vorlesungen über neuere Geometrie. Peano, Giuseppe (1889), Arithmetices principia, nova methodo exposita (Latin), excerpt reprinted in English stranslation as "The principles of arithmetic, presented by a new method", van Heijenoort 1976, pp. 83 97. Skolem, Thoralf (1920), "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen", Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse 6: 1–36. Tarski, Alfred (1948), A decision method for elementary algebra and geometry, Santa Monica, California: RAND Corporation Turing, Alan M Zermelo, Ernst (1908b), "Untersuchungen über die Grundlagen der Mengenlehre", Mathematische Annalen 65 (2): 261–281, doi:10.1007/BF01449999.