This accessible book gives a new detailed and elementary proof of the G�del incompleteness theorems and then presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities.

� propos de l?auteur

Gregory Chaitin is an Argentinian-American mathematician and computer scientist. The author of many books and scholarly papers, Chaitin proved the G�del-Chaitin incompleteness theorem and is the discoverer of the remarkable omega number, which shows that God plays dice in pure mathematics. Currently, he is attempting to create a mathematical theory of evolution and biological creativity, based on considering life as evolving software. He is a member of the International Academy of the Philosophy of Science and of the Brazilian Academy of Philosophy, and was awarded honorary doctorates from the University of Cordoba and the University of Maine. Chaitin is currently a visiting professor at the Federal University of Rio de Janeiro (UFRJ) in the program on Epistemology and History of Science and Technology (HCTE). He is also an honorary professor at the University of Buenos Aires. Newton da Costa is a Brazilian logician whose best known contributions have been in the realm of nonclassical logics. Da Costa developed paraconsistent logics, that is, logical systems that admit inner contradictions. Da Costa has wide-ranging interests, which go from foundational issues in the philosophy of science to physics (general relativity and quantum theory); besides his development of paraconsistent logics, he introduced the concept of quasi-truth to deal with mutually inconsistent scientific theories. Da Costa has a B. Sc. in civil engineering and a PhD in mathematics. He has visited several major universities (Stanford, Berkeley, Paris VII among others) and published about 200 scientific papers and several books on logic and the foundations of science. In 2009, he became a Professor Emeritus at Unicamp (Campinas, Brazil). Newton da Costa is a member of the Institut International de Philosophie, of the International Academy of the Philosophy of Science and of the Brazilian Academy of Philosophy. Francisco Antonio Doria is a Brazilian physicist. Doria is a Professor Emeritus at the Federal University of Rio de Janeiro, where he currently teaches economic theory at the graduate School of Engineering (UFRJ COPPE). Doria has a B. Sc. in chemical engineering and a PhD in mathematical physics. He has made contributions to the gauge field copy problem in quantum field theory and proved with Newton da Costa several incompleteness theorems in mathematics, physics and mathematical economics, including the undecidability of chaos theory. Doria is a member of the Brazilian Academy of Philosophy, was a Senior Fulbright Scholar at Stanford University, 1989-1990, and a visiting researcher at the mathematics department, University of Rochester.

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