Over the last few months, An Infinite Descent into Pure Mathematics has undergone a large overhaul, which I think will significantly improve its usefulness for readers. The result is version 0.2, which is a ‘preview’ of what will—hopefully this summer—be the first print edition.
What follows is a description of the major changes since version 0.1.
Both versions can be downloaded from the Archive page.
New structure
The most obvious difference between v0.1 and v0.2 is the order in which the sections appear and the overhaul of the material on logic and proofs.
Introductory material. The background material in the old §1.1 (Getting started) has been moved to a standalone chapter (new Ch 0) to separate it from the emphasis of new Ch 1 on logic. Its content is essentially unchanged, but I have a desire to expand it in order to be able to prove more interesting results in the early chapters of the book.
Logic and proofs. The material on proof techniques (old §1.2) and symbolic logic (old §2.1) has been almost entirely rewritten, although many of the examples remain the same as before. This material fills up new Ch 1 (Logical structure).
A lot of thought has gone into this change, with the guiding question being, why do we want students to learn about logic in the first place? The answer to that question is (or should be) that a solid understanding of the rules of logic assists with the creation, structure and wording of mathematical proofs. The updated material emphasises the connection between rules of logic and strategies for proof. Dealing with how the logical structure of a proposition informs the structure and wording of a proof will be the content of §A.2 (Writing a proof).
- New §1.1 (Propositional logic) and §1.2 (Variables and quantifiers) introduce symbolic logic side-by-side with direct proof techniques. The goal of these sections is to learn how to decompose a proposition in terms of its logical structure, and to use the logical structure of a proposition to inform a proof strategy. With this in mind, logical operators and quantifiers are now introduced in terms of the rules of inference that define them, which makes them more meaningful and informative for use in proofs than reducing them to Boolean algebra using truth tables.
- New §1.3 (Logical equivalence) introduces the notion of logical equivalence, now defined in terms of logical inference, as it should be. Truth tables are introduced now as a tool for proving logical equivalence, rather than as a means of defining logical equivalence. The emphasis of this section is on using logical equivalence to inform proof strategies.
Sets and functions. The material on functions was divided between old §2.3 and old §4.1—this material has been rearranged for added continuity and a more natural flow of content.
- New §2.1 (Sets and set operations) is essentially the same as old §2.2. Some of the material has been reworded or rearranged, new material on intervals and open sets in $\mathbb{R}$ has been added, and reference to absolute complements of sets has been removed.
- New §2.2 (Functions) is essentially the same as old §2.3, except the material on existence and uniqueness has been moved to new §1.2.
- New §2.3 (Injections and surjections) is essentially the same as the portion of old §4.1 up to, but excluding, the content on finite sets and counting. I felt guilty proving that every surjection has a right inverse without mentioning the axiom of choice (when, in fact, this assertion is equivalent to the axiom of choice), so this matter is now treated in this section.
Finite and infinite sets. The focus of old Ch 4 was on using injective and surjective functions to compare the sizes of sets, finite or infinite. With injections and surjections defined in new Ch 2, it seemed pertinent to give finite sets and infinite sets their own treatments in their own chapters. New Ch 3 (Finite sets) focuses on everything finite, and Ch 6 (Infinite sets) focuses on infinite, or rather ‘not-necessarily-finite’, sets.
- New §3.1 (The natural numbers) is the same in spirit as old §1.3 (Induction on the natural numbers), but has been improved in three main ways. First, definition by recursion is covered explicitly, rather than implicitly as it had been before. Second, a more careful treatment has been given of proof by strong induction where multiple base cases are required. Third, we cover the matter of Peano axioms and constructions of the natural numbers far more carefully than before. The order of material has been shuffled according to the new emphasis.
- New §3.2 (Finite sets) focuses on defining the term ‘finite’ and on proving criteria for finiteness and properties of finiteness. The questions of interest in new §3.2 are when a set is finite, rather than how many elements a finite set has. It combines material from old §4.1 (from ‘A first look at counting’ onwards) and old §4.2 (up to but excluding the combinatorial treatment of binomials and factorials).
- New §3.3 (Counting principles) is essentially the same as the latter part of old §4.2—the treatment of the addition and multiplication principles has been made less confusing and less messy.
- New §6.1 (Countable and uncountable sets) is essentially the same as old §4.3. It has been moved later in the book in order to allow more interesting examples to be given.
- Two new sections concerning infinite sets have been added, namely §6.2 (Cardinal arithmetic) and §6.3 (Ordinal arithmetic and the axiom of choice). When written, there will be a natural track through the book towards further study in mathematical logic and set theory: Ch 1 → Ch 2 → §3.1 & §3.2 → Ch 5 (at least §5.1 and §5.2) → Ch 6.
Number theory. The material on number theory (old Ch 3) was placed between two sections on functions (old §2.3 and §4.1). This was to provide temporary relief from the technical content on sets and functions, but it was unnatural from a mathematical point of view. Furthermore, some proofs in old Ch 3 used induction and counting principles. For these reasons, the number theory material appears in new Ch 4 (Number theory), after induction, functions and counting have been covered in their entirety. An instructor who is willing to skip some details in proofs could cover Ch 4 immediately after Ch 1.
Real analysis. The two sections on sequences and series have been combined into new §7.2 (Sequences and series), with room created for coverage of open sets and continuous functions in new §7.3 (Continuous functions).
Probability theory. Previously, only discrete probability spaces were covered, and old §7.3 (Expectation) was unnaturally short. The section on expectation of discrete random variables has been incorporated into new §8.2 (Discrete random variables), with a new §8.3 (Measure theory) focusing on defining the notion of a measure space and the more general notion of a probability space.
Additional topics. Old Ch 8 (Additional topics) has been deleted, since it was mostly unwritten anyway. Some of these topics have found their way into other sections, e.g. Boolean algebra in old/new §5.2, ordinal and cardinal numbers in new §6.2 and §6.3, and limits in new §7.3. The other topics will hopefully find their way into future versions of the book (it is an infinite descent, after all).
Appendices. The appendices have been slightly restructured, and the confusing material on theories and models has been deleted.
The approximate correspondences between new sections and old sections are summarised in the following table.
| New section/chapter | Old section(s)/chapter(s) |
| Ch 0 Getting started | §1.1 — content unchanged |
| §1.1 Propositional logic | §1.2 and §2.1 |
| §1.2 Variables and quantifiers | §1.2 and §2.1 |
| §1.3 Logical equivalence | §1.2 and §2.1 |
| §2.1 Sets and set operations | §2.2 — content unchanged |
| §2.2 Functions | §2.3 — content unchanged |
| §2.3 Injections and surjections | Beginning of §4.1 |
| §3.1 The natural numbers | §1.3 |
| §3.2 Finite sets | End of §4.1 and beginning of §4.2 |
| §3.3 Counting principles | End of §4.2 |
| Ch 4 Number theory | Ch 3 — content unchanged |
| Ch 5 Relations | Ch 5 — content unchanged |
| §6.1 Countable and uncountable sets | §4.3 — content unchanged |
| §6.2 Cardinal arithmetic | — |
| §6.3 Ordinal numbers and the axiom of choice | — |
| Ch 7 Real analysis | Ch 6 — content restructured |
| Ch 8 Discrete probability theory | Ch 7 — content restructured |
| – | Ch 8 Additional topics |
| Apx A Mathematical writing | Apx C incorporated as §A.3 |
| Apx B Miscellany | §B.2 |
| Apx C Hints to selected exercises | Apx A — content unchanged |
Other miscellaneous changes
- The default size for on-screen reading is now 6″ × 9″, which will (eventually) be the dimensions of the print version of the book. I have set a parameter to display the book in two-up format (in most PDF reader software), meaning less scrolling is required. The A4, US letter, phone and tablet versions remain available.
- Examples and exercises are now distinguished by colour; formerly they were both #007777 (for a reason I do not recall), but now exercises are #994400.
- The book now makes use of enumerated ‘Strategies’ rather than the ‘tips’ that were in the previous version. Each strategy is an interpretation of a definition, axiom or theorem in terms of how it can be used in a proof.
- The cleveref package has been implemented throughout, so now all references to definitions, theorems, sections, pages, and so on, are labelled automatically, and the label constitutes part of the hyperlink.
- The font has been changed to Times New Roman. I have nothing against Computer Modern, but for some reason it just looks better with Times.
- The ‘Proof’ label has been given more emphasis: it is now bold and on its own line for increased readability.
- End-of-chapter exercises are in the process of being added. If you are an instructor, I heavily recommend avoiding referring to these by number (e.g. Chapter 1 Q7) because the numbers will most certainly change—instead, I recommend copying out the question.
- An index of LaTeX notation has been added
Future updates
I am prioritising the following major changes for future versions of the book:
- Finishing the long-unfinished sections, notably §7.2 and §7.3 on the convergence of sequences and series. (It goes without saying that I want to fill in all the gaps that currently exist.)
- Including more exercises throughout, particularly end-of-chapter exercises.
- Writing the appendix on mathematical writing. I have the ideas but not the time!