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Incompleteness for higher-order arithmetic. An example based on Harrington’s principle. (English) Zbl 1457.03002

SpringerBriefs in Mathematics. Singapore: Springer (ISBN 978-981-13-9948-0/pbk; 978-981-13-9949-7/ebook). xiv, 122 p. (2019).
The book can be seen as a contribution to Harvey Friedman’s research program on concrete incompleteness for higher-order arithmetic. Three higher-order systems of arithmetic are introduced: Z\(_{2}\), Z\(_{3}\) and Z\(_{4}\) (set-theoretical axiomatic system for second-order, third-order and fourth-order, resp., arithmetic). A concrete mathematical theorem expressible in the language of Z\(_{2}\) which is neither provable in Z\(_{2}\) nor Z\(_{3}\) but provable in Z\(_{4}\) is provided. The crucial role in the investigation is played by the Martin-Harrington theorem and Harrington’s principle. The book is based on author’s doctoral dissertation and sequent work, however, it should be stressed that the book is a significant expansion and improvement of the dissertation. The author tried to make the book self-contained.

MSC:

03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03F35 Second- and higher-order arithmetic and fragments
03F40 Gödel numberings and issues of incompleteness
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