Yong Cheng Incompleteness for Higher-Order Arithmetic 9789811399480

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Автор: Yong Cheng
Название: Incompleteness for Higher-Order Arithmetic
ISBN: 9789811399480
Издательство: Springer
Классификация:

Математические основы

Математическая логика

ISBN-10: 9811399484
Обложка/Формат: Soft cover
Страницы: 122
Вес: 0.22 кг.
Дата издания: 2019
Серия: SpringerBriefs in Mathematics
Язык: English
Издание: 1st ed. 2019
Иллюстрации: 1 illustrations, black and white; xiv, 122 p. 1 illus.
Размер: 234 x 156 x 8
Читательская аудитория: Professional & vocational
Основная тема: Mathematics
Подзаголовок: An Example Based on Harrington’s Principle
Ссылка на Издательство: Link
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Поставляется из: Германии
Описание:

G?dels 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 Friedmans 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 “Harringtons 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 “Harringtons 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 Harringtons principle and its strengthening over second-order arithmetic and third-order arithmetic.

Дополнительное описание:

Introduction and Preliminary.- A minimal system.- The Boldface Martin-Harrington Theorem in Z2.- Strengthenings of Harrington’s Principle.- Forcing a model of Harrington’s Principle without reshaping.- The strong reflecting property for L-cardinals.<

Incompleteness for Higher-Order Arithmetic, Yong Cheng