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, RCA0, and WKL0. 0.7. Incompleteness in Nested Multiply Recursive Arithmetic, and Two Quantifier Arithmetic. 0.8. Incompleteness in Peano Arithmetic and ACA0. 0.9. Incompleteness in Predicative Analysis and ATR0. 0.10. Incompleteness in Iterated Inductive Definitions and 1 Π 1-CA0. 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. 2 The incompleteness phenomenon, in the sense understood today, was initiated 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 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. While it is true that Con(ZFC) can be stated entirely in terms of finite strings of symbols from a finite alphabet, when stated in this way, it is no longer natural in any mathematical sense. These considerations led to the informal working distinction between "mathematically natural" and "metamathematically natural". After the two incompleteness theorems, there remained the crucial question of whether there is a mathematically natural statement which is neither provable nor refutable in ZFC. This question had a potentially practical consequence. If the answer is no, then there is a clear sense in which mathematicians can forever be content to ignore the incompleteness phenomenon. However, if the answer is yes, 3 then there is a clear sense in which the incompleteness phenomenon can impact their work. Gödel addressed this question through his pioneering work on Cantor's Continuum Hypothesis (CH). CH states that every infinite set of real numbers is in one-one correspondence with either the integers or the real numbers. Gödel proved that ZFC does not suffice to refute CH. See [Go38], and [Go86-03], volume 2. That ZFC does not suffice to prove CH had to wait for the pioneering work of Paul J. Cohen, [Co63,64]. Also see [Je78,06]. Thus by the mid 1960s, a mathematically natural statement - the continuum hypothesis - was shown to be neither provable nor refutable in ZFC. Mathematical Incompleteness from ZFC was born. Yet mathematicians generally did not feel that CH was relevant to their work. This feeling of irrelevance went much deeper than just their particular research interests. There is a fundamental alienation of "questions like CH" from mathematical culture. Specifically, CH fundamentally involves a level and kind of generality that is entirely uncharacteristic of important and fruitful mathematical questions. Mathematicians will normally use general abstract machinery - when convenient - in the course of treating a relatively concrete problem. Witness the extensive use of general abstract machinery in Wiles' proof of Fermat's Last Theorem, and how much of this machinery can be removed (see [Mc10]). The general abstract machinery will be tamed if it causes its own difficulties or ceases to be convenient for various reasons. But the standards for objects of primary investigation of major interest are quite different. Sets of real numbers that play a role in mathematics as objects of primary investigation, are constructed in some fashion that is related to clear mathematical purposes. In virtually all cases, sets of real numbers appearing as objects of primary investigation, are Borel measurable (i.e., lie in the σ sigma generated by the open sets), and usually very low in the standard hierarchy of Borel measurable sets. 4 For Borel measurable sets of real numbers, the continuum hypothesis is a theorem, even in the following strong form: every infinite Borel measurable set of reals is in one-one correspondence with the integers, or in Borel one-one correspondence with the reals. See [Al16], [Hau16], and [Ke94], p. 83. This situation is typical of so many statements involving sets and functions in complete separable metric spaces. The Borel measurable forms are theorems, and have nothing to do with incompleteness. Furthermore, the great generality present in so many such statements is rather empty from the point of view of mathematical culture: there are virtually no mathematically interesting examples beyond Borel sets. There have been subsequent examples of ZFC incompleteness of less generality than arbitrary sets of reals. Most notably, involving the projective hierarchy of sets of reals, which is obtained by starting with Borel sets in several dimensions, and applying the operations of projection and complementation. Yet again, we see that the statements are decided in ZFC for Borel sets, and there are virtually no mathematically interesting examples that come under this generality beyond Borel sets. We take the view that Concrete Mathematical Incompleteness begins at the level of Borel measurable sets and functions on complete separable metric spaces. In section 0.3, we refine this to Mathematical statements concerning Borel measurable sets and functions of finite rank in and between complete separable metric spaces. We take the position that once we are discussing possibly very discontinuous functions between complete separable metric spaces, the Borel sets and functions of finite rank are not overly general - there are sufficient mathematically interesting examples of such reaching out to at least the first few finite levels. 5 In sections 0.11 - 0.13, incompleteness ranging from fragments of ZFC through ZFC and more are discussed in the setting of finite rank Borel sets and functions. In most cases, the incompleteness already starts kicking in at the first few finite ranks of the Borel hierarchy. However, Borel measurable sets and functions in complete separable metric spaces - even of low finite rank - is still substantially beyond what is considered normal for significant mathematical questions in the present mathematical culture. Incompleteness begins to become potentially noticeable when the examples live in discrete structures. Here by discrete structures, we mean finitely generated systems such as the ordered ring of integers, and the ordered field of rationals. We work with sets in and functions between discrete structures. Examples of incompleteness ranging from fragments of ZFC, to ZFC and beyond, are discussed in sections 0.5 - 0.10, and section 0.14. Boolean Relation Theory, the subject of this book, involves sets in and functions on the nonnegative integers. There is a brief account in section 0.14, and a detailed account in section 0.15. Some new developments that push Concrete Mathematical Incompleteness even further into the more immediately accessible and perfectly natural, are presented in section 0.14 without proof. The relevant manuscripts are under preparation. This Introduction concludes with a discussion of Concreteness in the realm of the Hilbert 1900 Problem List. This illustrates how the usual classification of mathematical statements used in mathematical logic (see the four displayed lists in section 0.3) relates to many contexts in core mathematics. The reader of this Introduction will see rather explicitly how the use of stronger and stronger fragments of ZFC, all the way through ZFC and extensions thereof by so called large cardinal hypotheses, supports proofs of more and more mathematically natural concrete statements. 6 In other words, this growing body of results shows rather explicitly what is to be gained by strengthening axiom systems for mathematics. 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, more strongly, that they prove only true arithmetic sentences - is an important one, but lies 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 briefly sketched.