There are a number of significant changes that I am planning to make in Version 1.0 of the book—over the next few months, these will begin to appear in the ‘Unstable download’ version.
This post contains a list of some of the planned (or already implemented) changes and the reasons for the changes, so that hopefully they don’t come as too much of a surprise when they are implemented.
- The strategy of proof by infinite descent will finally be addressed explicitly in Section 4.3 (Strong induction). After all, it is the proof strategy that the book is named after, so it feels wrong not to include it in the book.
- I am going through some of the more technical definitions and proofs in the book and attempting to rewrite them to make the core ideas clearer and the notation less grotesque. Some that come to mind include the technical results about finite sets in Section 6.1 (Finite sets) and Section 8.1 (Counting principles).
- The notation [n] will be re-defined to mean {k ∈ N | k < n} as opposed to {k ∈ N | 1 ≤ k ≤ n} — so, for example, [3] will now mean {0, 1, 2} instead of {1, 2, 3}. There are pros and cons to both definitions, but I have decided to make the change for two reasons. First and foremost, we use [n] to define finite sets, and the new definition generalises more naturally to reasoning about infinite sets and cardinal arithmetic. Second, several proofs are simplified by the equivalence between k ∈ [n] and k < n (where k ∈ N)—several proofs no longer require separate verifications that k ≥ 1, and several instances of ‘+1’ and ‘–1’ have been eliminated from proofs. This also has the beneficial effect that, for n > 0, the elements of [n] are precisely the remainders modulo n, which helps in Chapter 7 (Number theory).
The downside to this change is that it is slightly less intuitive for counting finite sets: we think of an enumeration function f : [n] → X as a way of counting the elements of X, and now the elements of X will be counted from 0 to n – 1 rather than from 1 to n. However, I am convinced that the technical benefits outweigh the intuitive costs, and I have faith in my readers’ ability to adapt to the new way of thinking.
- Chapter 0 (Getting Started) will have a big overhaul. The core goal of the chapter is the same as always, which is to introduce and get a feel for the number sets so that we have something to prove things about in Chapter 1 (Logical structure). The geometric interpretations of arithmetic operations may take a back seat (I haven’t decided yet), there will be more depth about sets (e.g. Chapter 0 will cover implied list notation, set-builder notation and intervals), and some more elementary material may be covered (notably the floor and ceiling operators on R).
- Chapters 7 (Number theory) and 8 (Enumerative combinatorics) will both be getting a big overhaul. Chapter 7, in particular, has barely been touched since I first wrote it in 2015, and is ripe for some improvement.
- Many more end-of-chapter exercises will be added to all chapters (including, notably, chapters that currently have no end-of-chapter exercises), and more hints will be provided to end-of-chapter exercises in Appendix C.
- Currently there are exercises within sections and at the end of each chapter. The within-section exercises serve one (or both) of two purposes: (a) to encourage readers to actively engage with the material and ensure they have grasped a concept before moving on, or (b) to avoid the need to provide the details of a proof that is elementary or similar to another proof that was already presented. On the other hand, the end-of-chapter exercises are intended for readers to deepen and strengthen their understanding, and for instructors using the textbook to assign as to their students. I will be making a more conscious effort to decide which exercises should go where (in-section vs. end-of-chapter), and I will provide solutions to most of the in-section exercises.
- Appendix B (Mathematical Miscellany) will be completed.
I will continue to update this list as I plan more changes to the book.