Difficulty of the book Mathematical Logic by Mileti

Question

I am currently working through the book Mathematical Logic by Mileti (online version) and just finished the second chapter (induction and recursion). I have to admit that it is quite challenging and whereas I think I can follow through, it really takes a lot of effort and time to thoroughly understand the ideas since they are so abstract for me. Has anyone worked through it and could let me know if the density and difficulty will continue in that rate or if one quickly gets used to it?

Thanks

Answer 1

As it happens, a friend of mine reviewed this book, and he asked for my feedback on the review, so I spent quite some time looking at it from a pedagogical viewpoint. (I don't think his review has appeared yet, or I'd link to it.)

It's one of the more difficult intro texts on the subject (though it has its strengths). Enderton’s A Mathematical Introduction to Logic or Leary’s A Friendly Introduction to Mathematical Logic would be easier going, I think.

We both questioned the value of the second chapter. Mileti provides a very abstract general framework for definitions by induction. Other books cover this material more concretely and intuitively. Typically you'll see "well-formed formula" of the sentential calculus defined via a few clauses, with maybe a page or two of discussion. Analogous definitions for the predicate calculus, free and bound variables, etc., appear later without much fuss and bother. The reader quickly gets used to the technique.

Mileti's approach allows him to quote general results for these specific cases. It provides a greater degree of rigor, and much later in the book it offers other dividends, but (IMO) his Chapter 2 should have been an appendix, or omitted altogether.

After this inauspicious start, the going does get smoother, but it remains less "user friendly" than most other intros. I particularly liked the wealth of applications to other branches of mathematics, more than you'll usually find in these books. On the other hand, if you don't have the necessary background in those fields (mostly algebra), this will not be a plus.