Intermediate Logic Text

Question

I have been self-studying logic and foundations of mathematics for some time. Unfortunately, I struggle to graduate to higher-level work. (I am particularly interested in model theory.) The problem is I cannot seem to find a set of texts that allows me to transition from 'baby logic' to more advanced work. There seems to be a large gap. I think for me the biggest problem is 1. the shear size and scope of the topic in general, and 2. a lack of notation consistency (where authors take much for granted). 3. There are a number of ways to formalize equivalent expressions, but not so easy to identify what is the same for a novice. I have perused numerous books and notes and T. Sider's seems to be on the right track. Can someone please recommend a/set of readings/texts that might help me 'get there?' Much thanks!

Answer 1

I'm not familiar with the T. Sider work you mention, but if you know some undergraduate logic and are looking to get into model theory at the graduate level, I recommend A Course in Model Theory by Tent and Ziegler (ISBN 9780521763240). That book presents the basics fairly cleanly, and also introduces many of the tools and ideas that underlie modern research (including stability, simplicity, and various ranks). Another possibility is Model Theory: An Introduction by Marker (ISBN 0387987606). Marker's book is more driven by algebraic examples, if that's to your taste.

I believe the above books (which should be available from your university library, or through inter-library loan) are good, accessible introductions to model theory. They are, as far as I recall, consistent about notation, and the notation they use is standard for the field. They do assume solid practical skills in advanced undergraduate mathematics, but I believe those skills are necessary to approach model theory at all. As far as specific prerequisite knowledge is concerned, both assume some basic knowledge of set theory (ordinal and cardinal arithmetic), and Marker assumes his reader knows a fair amount of algebra. If you find, reading either book, that you are consistently unable to follow the authors' reasoning, I recommend stepping back from model theory a bit and working more on general math background. In particular, learning some real analysis, e.g. from Rudin's Principles of Mathematical Analysis (ISBN 9780070542358), will help build skill in manipulating abstract mathematical ideas and in thinking like a mathematician.

Answer 2

You might take a look at my widely used Teach Yourself Logic Study Guide, which is aimed exactly at those self-studying logic, and wanting to move on from "baby logic" to more advanced stuff. It covers various areas of logic, including Model Theory, where it makes some recommendations of the more user-friendly books at various levels. You can freely download the 2015 edition here.