Описание: Kurt Goedel (1906-1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules.
Описание: Kurt Goedel (1906-1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules.
Автор: Yong Cheng Название: Incompleteness for Higher-Order Arithmetic ISBN: 9811399484 ISBN-13(EAN): 9789811399480 Издательство: Springer Рейтинг: Цена: 6986.00 р. Наличие на складе: Нет в наличии.
Описание:
G?del's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic.
This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp' is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp' and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.
Автор: Smullyan, Raymond M. Название: Godel`s incompleteness theorems ISBN: 0195046722 ISBN-13(EAN): 9780195046724 Издательство: Oxford Academ Цена: 46332.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: An introduction to the work of the mathematical logician Kurt Godel, which guides the reader through his Theorem of Undecidability and his theories on the completeness of logic, the incompleteness of numbers and the consistency of the axiom of choice.
Автор: Lindstr?m Название: Aspects of Incompleteness ISBN: 1107167922 ISBN-13(EAN): 9781107167926 Издательство: Cambridge Academ Рейтинг: Цена: 18216.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: This volume presents some of the main areas and results of general metamathematics. In addition to standard results of Goedel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, the book contains a thorough treatment of partial conservativity and degrees of interpretability. The method of arithmetization also plays an important role.
Описание: There`s Something About Godel is a lucid and accessible guide to Godel`s revolutionary Incompleteness Theorem , considered one of the most astounding argumentative sequences in the history of human thought. It is also an exploration of the most controversial alleged philosophical outcomes of the Theorem.
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