The revised Study Guide — preliminary instalment

As I’ve mentioned before, I have started work on revising/updating/extending/cutting-down the much-used Study Guide (Teach Yourself Logic as was, now retitled a bit more helpfully Beginning Mathematical Logic).

I’d thought about dropping the three-part structure. But I have decided, after some experimentation, to keep it. So after some preliminaries, Part I is on the core math logic curriculum. Part II (fairly short) looks sideways at some ways of deviating from/extending standard FOL. Part III follows up the topics of Part I at a more advanced level. So, for example, there is an introductory chapter on e.g. model theory in Part I, and then some suggestions about more advanced reading on model theory in another chapter in Part III. (Having one long chapter on model theory, one long chapter on arithmetic, etc. made for unwieldy and dauntingly long chapters, so that’s why it is back to the original plan.)

Over the next couple of weeks, I’ll be posting some early revised chapters from Part I, and I’ll very much be welcoming comments and suggestions (and corrections, of course) at this stage. Please, please, don’t hesitate to have your say (either using the comments boxes, or by email to peter_smith at logicmatters.net). A lot of students — possibly including your own students! — are downloading the Guide each month: if you think they are being led astray, now is the time to say!

Here then, for starters, are the Preface and a couple of preliminary chapters. Not terribly exciting, but much snappier than before. They will explain more about the structure and coverage of the Guide to those who don’t already know it. Next up, the long key chapter on FOL.

5 thoughts on “The revised Study Guide — preliminary instalment”

1. Concerning § 2.2: I understand that Button and Makinson are alternative textbooks to choose between, while Halmos is a book to read in addition to one of them (as is Russell, of course). Am I right? Maybe then should § 1.6 be slightly revised? (Or maybe I am being too pedantic…)

On page iii you explain what you mean when you mark a title with one star, but not what you mean by two stars.

1. Well, §1.6 applies to chapters in Part I, not this preliminary chapter. But I take the point.

Oops, that double star (surviving from an earlier version) should have been edited out! Thanks for spotting it.

2. General

I like the changes from the earlier revised chapters 1-9. The intended readership becomes clearer, sooner; the very useful ‘Strategies’ section appears much sooner (Ch1, p 4, rather than Ch3, p 12); and the separate introductions for philosophers and mathematicians have become one introduction that gets us to the informal set theory by p 7 rather than 14.

I like the way ‘set’ is now defined in (a) (i) on p 7, and I think it’s good that the item no longer talks of ‘identity’ (since it was a meaning / use of ‘identity’ that could be unfamiliar to many of the readers who need this chapter).

I also prefer the lighter-handed way the diagonal argument is now presented to the way the older version went into it in more detail in a footnote.

OTOH, I still think the section on virtual classes is too much, and that it could confuse and cast unnecessary doubt more than enlighten.

Pluralist “some”

Example of where plural-speak “some” aligns with ordinary English:

Some cats went into a bar.

Examples where it doesn’t:

Some cats are friendly.
Some cats are friendly if and only if you feed them.

Halmos, Naive Set Theory

Plus:

* short
* inexpensive (though it didn’t used to be)
* written in a style that’s unusual for a maths text and that some people like
* of historical / sociological interest

Minus:

* text-dense
* no cumulative hierarchy (V)
* no mention of order types
* almost no exercises (as normally understood)
* stops just as things are starting to get interesting
* shows its age
– “All that is known for sure is that the continuum hypothesis is consistent with the axioms of set theory”
– The Axiom of separation is called Axiom of specification and, we’re told, is “often referred to by its German name Aussonderungsaxiom”
– Axiom of substitution rather than replacement

* It’s almost, but not quite, ZFC, because it doesn’t include the axiom of foundation.

It’s written, quite explicitly, for students who aren’t interested in set theory, who are, instead, as his Preface puts it, “anxious to study groups, or integrals, or manifolds”; and “The student’s task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities … general set theory is pretty trivial stuff really”. I don’t think textbook authors should talk down their subject; I think they should treat the reader as someone who might actually find it interesting.

Compare Halmos to Judith Roitman in the Preface to her Introduction to Modern Set Theory:

When, in early adolescence, I first saw the proof that the real numbers were uncountable, I was hooked. I didn’t quite know on what, but I treasured that proof, would run it over in my mind, and was amazed that the rest of the world didn’t share my enthusiasm. Much later, learning that set theorists could actually prove some basic mathematical questions to be unanswerable, and that large infinite numbers could effect the structure of the reals — the number line familiar to all of us from the early grades — I was even more astonished that the world did not beat a path to the set theorist’s door.

1. Thanks for the positive general comments. As to the section of virtual classes, I’ve had positive reactions from others: it’s a judgement call how much (if anything to say).

On the Halmos book. Maybe I am guilty here of the same sin that I accuse others of when e.g. they stick to Mendelson — a sentimental attachment to an old friend! I should revisit it. (Though some points you have as strikes against it “almost no exercises (as normally understood), stops just as things are starting to get interesting” aren’t so negative in its intended role here as preliminary, scene-setting, reading.)

1. The virtual classes section

I wonder whether the positive reactions are just because the people agree with what the section says — or whether they also think that the issue is so significant and important that it had better be explained, at length, very soon after sets are first introduced, and in the most introductory part of the guide (in a chapter that seemed to be for students who “aren’t already familiar” with the “absolute basics”).

I think it can be more confusing than helpful, especially since one of the recommended books — Makinson’s — uses “class” differently: as a synonym for “set”, because he thinks “the phrase ‘set of sets’ tends to be difficult for the mind to handle” (p 105) and so will “speak synonymously of a collection, or class of sets.” (p 5)

I’m not saying remove the section completely, though, if you think it’s important to say something. I just think it would be better if it were cut back to one or two paragraphs. That would tell the reader there is an issue, and perhaps make them a bit prepared for a class / set distinction, if they should encounter one later on, without filling their head with potentially confusing details that aren’t important at this stage.

Halmos

I’m surprised by what you say about exercises. I’d have thought that exercises would be seen as especially useful in introductory reading, in a self-study / teach-yourself context.

There used to be a book of exercises that was meant to be used with Naive Set TheoryExercises in Set Theory by L. E. Sigler — but it has long been out of print.

However, while looking around yesterday, I found that someone (George Mplikas) has written solutions to what looks like all of the things labelled “exercises” in NST, and to some that aren’t:

https://raw.githubusercontent.com/gblikas/set-theory-solutions-manual/master/SolutionsManual.pdf

I haven’t looked at it in detail to see whether the solutions are correct, etc, but it’s possible that people who take up NST would find it useful.

BTW, if we take what Halmos says about exercises seriously —

There are only a few exercises, officially so labelled, but, in fact, most of the book is nothing but a long chain of exercises with hints.

— it sounds an awful lot like there’s much that he doesn’t explain clearly, instead providing only hints.

And indeed I think the book is often harder going (and requires more mathematical experience / maturity) than its conversational style initially suggests.

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