GODEL PROOF NAGEL NEWMAN PDF

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.

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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. This important result states that any first-order theorem which is true in all models of a theory must be logically deducible from that theory, and vice versa for example, in abstract algebra any result which is true for all groups, must be deducible from the group axioms.

This is, of course, not an absolute apriori proof of consistency, as originally dreamt by Hilbert, but it is quite an important consideration that should not be forgotten either. This is actually what happened historically, when more sophisticated theories such as ZFC developed out of the naive set theories initially proposed by set theorists. Anyway, going back to this remarkable book, I think that it is one of the best not-fully-technical available treatments of these seminal theorems: it is very highly recommended to any reader provided with some basic background knowledge of logic and set theory, and willing to explore these theorems to some good level of detail.

It is not complete and a bit dated in parts, but an excellent treatise nevertheless, fully deserving a 5-star rating.

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Gödel's Proof

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.

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