Tim Button’s Set Theory: A Open Introduction

Tim Button wrote up his Cambridge lecture notes for a course on set theory for philosophers, previously taught by Michael Potter and then Luca Incurvati, as Open Set Theory which he then contributed to the Open Logic Project, and the resulting Set Theory: An Open Introduction can now be downloaded from this page on the Project’s site. The result, as you would expect from this author, is very good. It makes an excellent, and free, alternative to e.g. Enderton’s famous introductory book. It is particularly clear in marking off the initial informal (naive) development of the theory of sets, cardinals, ordinals etc. from the ensuing elaboration of an official iterative conception of the hierarchy and the formal development of an axiomatization of ZFC. It is perhaps just occasionally uneven in level; but when I do a half-yearly update of the Teach Yourself Logic Study Guide in a few weeks time, this will get promoted to a top recommendation.

I’ll be checking out again more of Open Logic’s offerings, for this project seems to have developed very well. In particular, I’ll also be recommending Richard Zach’s Incompleteness and Computability, which you can download here. More in TYL 2020.5. 

5 thoughts on “Tim Button’s Set Theory: A Open Introduction”

  1. I have taught both propositional logic and first-order logic using *forall x: Calgary*. I find it very clear and easy to follow. I believe it could be a perfect self-study text. I am not sure however if the other texts in the project could be used without a teacher.

    1. I like forall x. And in fact, the new edition of IFL2 (due out very soon) takes a similar approach with a very similar Fitch-style proof system. Surprise, surprise, I’d recommend IFL2 over forall x for self-study! — if only because I have the space to cover the same things singificantly more slowly with more examples and add more philosophical asides.

      1. Unfortunately I have not read your book entirely yet. However, I went through the chapters on the tree method that you uploaded few months back. I fond it quite interesting and very easy to follow.

        One of the things that I like about forallx is that it introduces the use/mention distinction and is quite careful about the distinction throughout the text. It might be a little hard a distinction for students when they encounter it for the first time, but it is very helpful to have it in mind. (it also has a solution booklet!)

  2. I’ve looked at a fair amount of the book and have also looked back at two classic texts — Halmos and Enderton — to see how they handle things.

    I like most of what I’ve seen so far, and I even prefer it to Enderton’s book in some ways (though not overall). There are also some things I don’t like.

    I think it puts too much emphasis on its informal presentation being “naive”. Most of it is not especially naive, and some things it calls “naive” could be done the same way using axiomatic set theory. (Enderton’s book applies “naive” to set theory very rarely — only in one paragraph, on p 11 — and calls it “a terminology that does not hide its bias”.)

    I also think that, by running “informal” and “naive” together, it conflates two different meanings of “naive”. One of them does indeed go with “informal”, with a connotation that it’s also intuitive and pre-axiomatic; the other is specifically about unrestricted (“naive”) comprehension, the logical conception, and the sort of set theory Frege developed. Informal set theory does not have to be “naive” in that second sense, but that’s not quite how this book makes it seem. And it’s not necessary to abandon being “naive” in the first sense in order to avoid falling into Russell’s Paradox. Again, that’s not how this book makes it seem. Instead, §2.6 says that to “set up a set theory which avoids falling into Russell’s Paradox, … we would need to lay down axioms which give us very precise conditions for stating when sets exist (and when they don’t).”

    Button begins with an interesting discussion of mythologised history — it’s one of the things I like, though when it says “the history of mathematics was largely written by the victors”, it’s not entirely clear who the victors are meant to be.

    However, when the two meanings of naive are run together, we get something very like a mythologised history in which it seems that there’s only one informal / intuitive / pre-axiomatic approach, and that it involves unrestricted comprehension. Who are the “victors” who would write history that way? Not the advocates of the combinatorial or iterative conceptions, nor those who think Cantor (or set theory in general) should not share the blame for Frege’s mistake.

    As Godel pointed out in “What is Cantor’s continuum problem?”, the iterative conception can be used “naively” in the first sense, and “has never led to any antinomy whatsoever”.

    Enderton’s book is an example: it is largely based on the iterative conception and presents it informally. The preface says (p xii), “The hierarchical view of sets, constructed by transfinite iteration of the power set operation, is adopted from the start.” And the cumulative V hierarchy is indeed introduced very early on: p 7, complete with the sort of V-shaped diagram that does not appear in Button’s book until p 114, and even then showing only the levels indexed by finite ordinals. (Halmos does not describe the Vs at all and has no such diagram.)

    This allows Enderton to say on p 8

    A fundamental principle is the following: Every set appears somewhere in this hierarchy. That is, for every set a there is some α with a ∈ Vα+1 That is what the sets are; they are the members of the levels of our hierarchy.

    This is not dumped on the reader. There’s an earlier section in Chapter 1, Baby Set Theory, that describes the elementary set theory many readers would already have encountered: the empty set, union, intersection, powerset, … leading to a discussion of “some dangers inherent in the abstraction method” (comprehension, though Enderton quite reasonably does not use that name). Russell’s paradox is presented as an example of something we need to be careful about, not as a fall from paradise, and Enderton says such “disasters will be blocked in precise ways in our axiomatic treatment, and less formally in our nonaxiomatic treatment.”

    This is followed by the section that introduces the cumulative hierarchy: Sets — an Informal View. Transfinite levels are introduced, and we’re told

    A better explanation of the “forever” idea must be delayed until we discuss (in Chapter 7) the “numbers” being used as subscripts in the preceding paragraphs. These are the so-called “ordinal numbers.” The ordinal numbers begin with 0, 1,2, …; then there is the infinite number ω, then ω+1, ω+2, …;and this goes on “forever.”

    There are some simple exercises about the hierarchy to help the reader get used to the idea.

    Enderton does say that the paradoxes “forced the development of axiomatic set theory, by showing that certain assumptions, apparently plausible, were inconsistent and hence totally untenable” (p 11). However, he also points out that

    Even without the foundational crises posed by the paradoxes of naive set theory, the axiomatic approach would have been developed to cope with later controversy over the truth or falsity of certain principles, such as the axiom of choice (Chapter 6)

    And he sees Russell’s paradox as especially relevant to Frege’s approach: ” the simplicity and the directness of Russell’s paradox seemed to destroy utterly the attempt to base mathematics on the sort of set theory that Frege had proposed.” (p 18)

    Halmos (and Enderton)

    Despite calling his book Naive Set Theory, what Paul Halmos presented was ZFC. Here, from his preface, is how he saw the terrain:

    In set theory “naive” and “axiomatic” are contrasting words. The present treatment might best be described as axiomatic set theory from the naive point of view. It is axiomatic in that some axioms for set theory are stated and used as the basis of all subsequent proofs. It is naive in that the language and notation are those of ordinary informal (but formalizable) mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study. …

    He treats Russell’s paradox in a very matter-of-fact way as “instructive application of the axiom of specification” that shows there’s no set of all sets. Then:

    In older (pre-axiomatic) approaches to set theory, the existence of a universe was taken for granted, and the argument in the preceding paragraph was known as the Russell paradox. The moral is that it is impossible, especially in mathematics, to get something for nothing. To specify a set, it is not enough to pronounce some magic words (which may form a sentence such as “x ∉ x“) ; it is necessary also to have at hand a set to whose elements the magic words apply.

    That’s a far cry from the moral Quine seemed to draw, “Common sense is bankrupt”!

    A peculiarity of Enderton’s Elements of Set Theory is that it gives the reader a choice about how ‘axiomatically’ to read it. The specifically axiomatic material is even marked by a line (like a change bar) down the margin “to allow a user to deemphasize the axiomatic material, or even to omit it entirely.” (p xi)

    (I wonder whether this was at least partly because his book had to compete with Halmos’s, and either Enderton or his publisher did not want to make it seem it was only for people who wanted an axiomatic approach.)

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