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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. 0.17. Concreteness in the Hilbert Problem List. This Introduction sets the stage for the new advances in Concrete Mathematical Incompleteness presented in this book. The remainder of this book can be read without relying on this Introduction. However, we advise the reader to peruse this Introduction in order to gain familiarity with the larger context. Readers can proceed immediately to the overview of the contents of the book by first reading the brief account in section 0.14C, and then 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.
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2 The incompletess phenomenon, in the sense understood today, was initiatied by Kurt Gödel with his first incompleteness theorem, where he essentially established that there are sentences which cannot be proved or refuted using the usual axioms and rules of inference for mathematics, ZFC (assuming ZFC is free of contradiction). See [Go31], and [Go86-03], volume 1. Gödel also established in [Go31] that this gap is not repairable, in the sense that if ZFC is extended by finitely many new axioms (or axiom schemes), then the same gap remains (assuming the the extended system is free of contradiction). With his second incompleteness theorem, Gödel gave a critical example of this incompleteness. He showed that the statement Con(ZFC) = "ZFC is free of contradiction" is neither provable nor refutable in ZFC (assuming ZFC is legitimate in the sense that it proves only true statements in the ring of integers). Again, see [Go31], and [Go86-03], volume 1. Although Con(ZFC) is a natural statement concerning the axiomatization of abstract set theory, it does not represent a natural statement in the standard subject matter of mathematics.
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