Cheng, Yong 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. Reviewer: Roman Murawski (Poznań) Cited in 2 Documents 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 Keywords:incompleteness; higher-order arithmetic; Harrington’s principle; Martin-Harrington theorem PDFBibTeX XMLCite \textit{Y. Cheng}, Incompleteness for higher-order arithmetic. An example based on Harrington's principle. Singapore: Springer (2019; Zbl 1457.03002) Full Text: DOI