Being relatively short, this book does not expand on the important correspondences and similarities with the concepts of computability originally introduced by Turing in theory of computability, particularly in the theory of recursive functions, there is a fundamental theorem stating that there are semi-decidable sets sets which can be effectively generated , that are not fully decidable. As expressed beautifully by Chaitin, uncomputability is the deeper reason for incompleteness. And it is precisely by using this fundamental result that Godel could demonstrate his celebrated theorems. Given a formal system such as PA or ZFC, the relationship between the axioms and the theorems of the theory is perfectly mechanical and deterministic, and in theory recursively enumerable by a computer program. Metamathematical arguments establishing the consistency of formal systems such as ZFC have been devised not just by Gentzen, but also by other researchers. For example, we can prove the consistency of ZFC by assuming that there is an inaccessible cardinal.
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In he graduated from the City College of New York, where he had studied under Morris Cohen, with whom he later collaborated to coauthor the highly successful textbook, An Introduction to Logic and Scientific Method Pursuing graduate studies at Columbia University, he received his Ph. In he joined the faculty of Rockefeller University. Nagel was one of the leaders in the movement of logical empiricism, conjoining Viennese positivism with indigenous American naturalism and pragmatism. In he published in the Journal of Philosophy the article "Impressions and Appraisals of Analytic Philosophy," one of the earliest sympathetic accounts of the works of Ludwig Wittgenstein, Moritz Schlick, and Rudolf Carnap intended for an American audience. Nagel was esteemed for his lucid exposition of the most recondite matters in logic, mathematics, and natural science, published in essays and book reviews for professional journals, scientific periodicals, and literary reviews. He also wrote a monograph, Principles of the Theory of Probability which appeared in the International Encyclopedia of Unified Science.