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1 INTRODUCTION CONCRETE MATHEMATICAL INCOMPLETENESS 0.1. General Incompleteness. 0.2. Some Basic Completeness. 0.3. Abstract and Concrete Mathematical Incompleteness. 0.4. Reverse Mathematics. 0.5. Incompleteness in Exponential Function Arithmetic. 0.6. Incompleteness in Primitive Recursive Arithmetic, Single Quantifier Arithmetic, RCA 0 , and WKL 0 . 0.7. Incompleteness in Nested Multiply Recursive Arithmetic, and Two Quantifier Arithmetic. 0.8. Incompleteness in Peano Arithmetic and ACA 0 . 0.9. Incompleteness in Predicative Analysis and ATR 0 . 0.10. Incompleteness in Iterated Inductive Definitions and Π 1 1 -CA 0 . 0.11. Incompleteness in Second Order Arithmetic and ZFC\P. 0.12. Incompleteness in Russell Type Theory and Zermelo Set Theory. 0.13. Incompleteness in ZFC using Borel Functions. 0.14. Incompleteness in ZFC using Discrete Structures. 0.15. Detailed overview of book contents. 0.16. Some Open problems. This Introduction sets the stage for the new advances in Concrete Mathematical Incompleteness presented in this book. Readers can proceed immediately to the overview of the contents of the book by reading the brief overview in section 0.14C, and the fully detailed overview in section 0.15. These are self contained and do not rely on the rest of the Introduction. In this Introduction, we give a general overview of what is known concerning Incompleteness, with particular emphasis on Concrete Mathematical Incompleteness. The emphasis will be on the discussion of examples of concrete mathematical theorems - in the sense discussed in section 0.3 - which can be proved only by using unexpectedly strong axioms. This growing body of results shows rather explicitly what is to be gained by strengthening axiom systems for mathematics.
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2 Of course, there is an even greater loss realized by strengthening a consistent axiom system to an inconsistent one. The issue of why we believe, or why we should believe, that the relevant axiom systems used in this book are consistent - or even that they prove only true arithmetic sentences - is an important one, and must lie beyond the scope of this book. Since this Introduction is to be viewed as clarifying background material for the six Chapters, many of the proofs are sketchy, and in many cases, we have only provided sketches of the ideas behind the proofs. We have also included results from the folklore, results that can be easily gleaned from the literature, and results of ours that we intend to publish elsewhere. We have provided many relevant references. No results presented in this Introduction will be used subsequently in the book (except for Gödel's second incompleteness theorem). 0.1. General Incompleteness. General Incompleteness was initiated by Gödel's landmark First and Second Incompleteness Theorems, which apply to very general formal systems. The original reference is [Go31].
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