Gödel'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.

Autorentext

Prof. Sheng Zhang received his B.E. degree in Energy and Environment Application Engineering from Zhejiang University in 2014 and his Ph.D. degree in Building Services from City University of Hong Kong in 2020. He served as a postdoc at City University of Hong Kong for one year and has been working at Xi’an Jiaotong University as an associate professor since 2021. Dr. Zhang’s research interest focuses on indoor environment quality-oriented low-carbon building technology, including 1) demand theories of thermal comfort and indoor air quality for both non-pandemic and pandemic scenarios; 2) systematic ventilation performance indices; 3) advanced air distribution and control methods; 4) air and ground sources-based renewable heating and cooling energy systems. Dr. Zhang has published more than 80 SCI papers in JCR Q1 journals (as first/corresponding author for more than 60 SCI papers) and received more than 10 research projects. Ms. Jinghua Jiang received the B.E. degree from Hebei University of Technology in 2020 and the M.E. degree from Xi’an Jiaotong University in 2023. She is a Ph.D. student at Xi’an Jiaotong University. Ms. Jinghua Jiang’s research interests include advanced air distribution and geothermal energy for building heating and cooling. She has published 8 SCI papers, received the First Prize of Scientific and Technological Research Achievements of Higher Education Institutions in Shaanxi Province, and received the Award of Excellent Graduate Student of Xi’an Jiaotong University. Prof. Yong Cheng received his B.E. and M.E. degrees in Built Environment and Energy Application Engineering from Hunan University in 2008 and Heating, Gas Supply, Ventilation, and Air Conditioning Engineering from Tongji University in 2011, respectively, and his Ph.D. degree in Building Services from City University of Hong Kong in 2015. Prof. Cheng has been working at Chongqing University since 2016. Prof. Cheng focuses on the research of Building Ventilation. His recent research interests include advanced air distribution, human thermal comfort, contaminant dispersion, building energy efficiency, etc. Prof. Cheng has published more than 80 peer-reviewed academic papers in international journals and conferences, and has received more than 10 research projects. Prof. Zhang Lin is the Chair Professor in Built Environment at City University of Hong Kong. Prof. Lin received his B.E. degree in Air Conditioning from Tsinghua University in 1983 and his Ph.D. degree in Processing and Environmental Technology from Massey University in 1995. Prof. Lin’s research interests are Built Environment and Renewable Energy Systems, including advanced air distribution, indoor and outdoor thermal comfort, indoor air quality, solar energy systems, etc. Prof. Lin has published more than 200 SCI papers and is among the world’s Top 2% scientists. He is a member of the American Society of Heating, Refrigeration, and Air Conditioning Engineers, the Institution of Engineers of Australia, the Hong Kong Institution of Engineers, the Extended Building Committee, the Building Department, the Hong Kong SAR government, etc. Prof. Lin is an editorial board member of the international journals of Building and Environment, and Building Simulation.

Klappentext
The book examines the following foundation question: are all theorems in classic mathematics which are expressible in second order arithmetic provable in second order arithmetic? In this book, the author gives a counterexample for this question and isolates this counterexample from Martin-Harrington theorem in set theory. It shows that the statement Harrington's principle implies zero sharp is not provable in second order arithmetic. The book also examines what is the minimal system in higher order arithmetic to show that Harrington's principle implies zero sharp and the large cardinal strength of Harrington's principle and its strengthening over second and third order arithmetic.

Inhalt
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.