Elementary Set Theory


20110404(Smbc. Here are the links)

I finished this book earlier today. This is a really hard book to blog, and I ended up concluding that I should just cover the material here by adding some links to relevant wiki articles dealing with stuff also covered in the book. It’s presumably far from obvious to someone who has not read the book how the different links are related to each other, but adding details about this as well would mean rewriting the book. Consider the articles samples of the kind of stuff covered in the book.

It should be noted that the book has very few illustrations and a lot of (/nothing but) formulas, theorems, definitions, and examples. Lots of proofs, naturally. I must admit I found the second half of the book unnecessarily hard to follow because it unlike the first half included no images and illustrations at all (like the ones in the various wiki articles below) – you don’t need an illustration on every second page to follow this stuff, but in my opinion you do need some occasionally to help you imagine what’s actually going on; this stuff is quite abstract enough to begin with without removing all ‘visual aids’. Some of the stuff covered in the book was review, but a lot of the stuff covered was new to me and I did have a few ‘huh, I never really realized you could think about [X] that way‘-experiences along the way. One problem making me want to read the book was also that although I’ve encountered some of the concepts introduced along the way – to give an example, the axiom of choice and various properties of sets, such as countability, came up during a micro course last year – I’ve never really read a book about that stuff so it’s sort of felt disconnected and confusing to some extent because I’ve been missing the big picture. It was of course too much to ask for to assume that this book would give me the big picture, but it’s certainly helped a bit to read this book. I incidentally did not find it too surprising that the stuff in this book sort of overlaps a bit with some stuff I’ve previously encountered in micro; in a book like this one there are a lot of thoughts and ideas about how to set up a mathematical system, and you start out at a very simple level and then gradually add more details along the way when you encounter problems or situations which cannot be dealt with with the tools/theorems/axioms already at your disposal – this to me seems to be a conceptually similar approach to numbers and mathematics as the approach to modelling preferences and behaviour which is applied in e.g. Mas-Colell et al.‘s microeconomics textbook, so it makes sense that some similar questions and considerations pop up. A related point is that some of the problems the two analytical approaches are trying to solve are really quite similar; a microeconomist wants to know how best to define preferences and then use the relevant definitions to compare people’s preferences, e.g. by trying to order them in various ways, and he wants to understand which ways of mathematically manipulating the terms used are permissible and which are not – activities which, at least from a certain point of view, are not that different from those of set theorists such as those who wrote this book.

I decided not to rate the book in part because I don’t think it’s really fair to rate a math book when you’re not going to have a go at the exercises. I wanted an overview, I don’t care enough about this stuff to dive in deep and start proving stuff on my own. I was as mentioned annoyed by the lack of illustrations in the second part and occasionally it was really infuriating to encounter the standard ‘…and so it’s obvious that…’ comments which you can find in most math textbooks, but I don’t know how much these things should really matter. Another reason why I decided not to rate the book is that I’ve been suffering from some pretty bad noise pollution over the last days, which has meant that I’ve occasionally had a really hard time concentrating on the mathematics (‘if I’d just had some peace and quiet I might have liked the book much better than I did, and I might have done some of the exercises as well’, is my working hypothesis).

Okay, as mentioned I’ve added some relevant links below – check them out if you want to know more about what this book’s about and what kind of mathematics is covered in there. I haven’t read all of those links, but I’ve read about those things by reading the book. I should note, in case it’s not clear from the links, that some of this stuff is quite weird.

Set theory.
Bijection, injection and surjection.
Indexed family.
Power set.
Empty set.
Family of sets.
Axiom of choice.
Successor function.
Axiom of infinity.
Peano axioms.
Equivalence class.
Equivalence relation.
Schröder–Bernstein theorem.
Cantor’s theorem.
Cantor’s diagonal argument.
Order theory.
Transfinite induction.
Zorn’s Lemma.
Ordinal number.
Cardinal number.
Cantor’s continuum hypothesis.


November 4, 2014 - Posted by | books, mathematics

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