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Mathematical logic

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Proof theory and constructive mathematics: Added links
Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as the [[Gödel–Gentzen negative translation]] show that it is possible to embed (or ''translate'') classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs.
Recent developments in proof theory include the study of [[proof mining]] by [[Ulrich Kohlenbach]] and the study of [[proof-theoretic ordinal]]s by Michael Rathjen.
"Mathematical logic has been successfully applied not only to mathematics and its foundations ([[Gottlob Frege|G. Frege]], [[Bertrand Russell|B. Russell]], [[David Hilbert|D. Hilbert]], [[Paul Bernays|P. Bernays]], [[Heinrich Scholz|H. Scholz]], [[Rudolf Carnap|R. Carnap]], [[Stanislaw Lesniewski|S. Lesniewski]], [[Thoralf Skolem|T. Skolem]]), but also to physics (R. Carnap, A. Dittrich, B. Russell, [[Claude Shannon|C. E. Shannon]], [[Alfred North Whitehead|A. N. Whitehead]], [[Hans Reichenbach|H. Reichenbach]], P. Fevrier), to biology ([[Joseph Henry Woodger|J. H. Woodger]], [[Alfred Tarski|A. Tarski]]), to psychology ([[Frederic Fitch|F. B. Fitch]], [[Carl Gustav Hempel|C. G. Hempel]]), to law and morals ([[Karl Menger|K. Menger]], U. Klug, P. Oppenheim), to economics ([[John von Neumann|J. Neumann]], [[Oskar Morgenstern|O. Morgenstern]]), to practical questions ([[Edmund Berkeley|E. C. Berkeley]], E. Stamm), and even to metaphysics (J. [Jan] Salamucha,<ref>"Jan Salamucha", .</ref> H. Scholz, [[Jozef Maria Bochenski|J. M. Bochenski]]). Its applications to the history of logic have​ proven extremely fruitful ([[Jan Lukasiewicz|J. Lukasiewicz]], H. Scholz, [[Benson Mates|B. Mates]], A. Becker, [[Ernest Addison Moody|E. Moody]], J. Salamucha, K. Duerr, Z. Jordan, [[Philotheus Boehner|P. Boehner]], J. M. Bochenski, S. [Stanislaw] T. Schayer,<ref>"Stanislaw Schayer", .</ref> [[Daniel H. H. Ingalls Sr.|D. Ingalls]])."<ref>Jozef Maria Bochenski, ''A Precis of Mathematical Logic'', rev. and trans., Albert Menne, ed. and trans., Otto Bird, Dordrecht, South Holland: Reidel, Sec. 0.3, p. 2.</ref>
==Connections with computer science==