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!
-
$\begingroup$ I don't have quite enough reputation to make the edit, but I suggest tagging "model-theory" and "reference-request", in order to get more responses. $\endgroup$– user231101Sep 24, 2015 at 0:01
-
$\begingroup$ Thanks Mike. Theodore Sider's "Logic for Philosophy". I have been hitting the study from the philosophy end as opposed to the mathematical end. I have looked at the two books you mentioned. I am just concerned about purchasing expensive books and the learning stopping at page 25 or so. $\endgroup$– semwilSep 24, 2015 at 0:10
-
$\begingroup$ Are you at a university? You can probably get any relevant books from the library, or inter-library loan if your school doesn't have them. $\endgroup$– user231101Sep 24, 2015 at 0:14
-
$\begingroup$ Yes, I have access to university library. I'm looking for the "secret books" if you will. There are a set of books in any given discipline who purpose is to actually demonstrate explicitly the material, given some explicit prior knowledge. They are written at the intended level as claimed. E.g. How often do we come across texts that, if I understood the level the author is writing at, then I wouldn't be reading this book. I have found some of these books/articles/notes, but they are rare and obscure in some cases. All I really find is baby logic and then grand master specialized logic. $\endgroup$– semwilSep 24, 2015 at 0:32
-
$\begingroup$ What is your math background? The missing secret ingredient may be the experience from a strong undergrad-level math education. Model theory isn't really approachable (regardless of how it's explained) without it. A very deep background in technical philosophy may help, but model theory is ultimately a branch of mathematics. Intuition, motivation, and examples come from (advanced) algebra, geometry, analysis, etc. There is a reason it's usually only taught as a graduate-level course in math departments. $\endgroup$– user231101Sep 24, 2015 at 0:53
2 Answers
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.
-
$\begingroup$ You should disclose that you are the author. $\endgroup$– user231101Sep 24, 2015 at 14:41
-
$\begingroup$ It's hardly a state secret ;-) But I've added "my" ... $\endgroup$ Sep 24, 2015 at 15:18
-
$\begingroup$ Thank you Peter. I recently came across your web and I intend on using that material. I am also looking at Oxford reading list. Thank you for your work and making it available! $\endgroup$– semwilSep 25, 2015 at 1:24
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.
-
$\begingroup$ I'd have thought both Marker and Tent & Ziegler (good though they are) are on the far side of the gap which the OP wants bridged. Sider's book is pretty noddy (and not much good), and it is a huge ask to jump from that all the way to those two texts. $\endgroup$ Sep 24, 2015 at 14:39
-
$\begingroup$ @PeterSmith I believe our disagreement comes from a difference in perspective on logic between philosophers and mathematicians. Philosophy isn't specified in the question itself, and I do believe my answer is good advice for a mathematician. To a budding mathematician, if there is a gap to bridge before reading M. or T.&Z., it is a gap of mathematical maturity. My answer addresses that possibility. I don't know how to prepare a philosopher to study model theory. To be honest, I don't even know what "model theory research" would mean in a philosophy department (or if such a thing even exists). $\endgroup$– user231101Sep 24, 2015 at 15:01
-
$\begingroup$ I am not really disagreeing with your recommendations as eventual destinations -- as I said, they are good books. But the OP is a self-professed "novice", and even novice mathematicians aren't going to find those avowedly graduate-level texts an easy route. If the OP is at Sider's level, then s/he will very likely want something like Manzano's stand-alone book, or the later chapters in logic texts that cover some elementary model theory $\endgroup$ Sep 24, 2015 at 15:17
-
$\begingroup$ Thank you both! I didn't mean to over emphasis model theory per se. It just seems to be where my intuition is guiding me. I do need mathematical maturity. I looked at the recommended books. I think they are a bit dense for me to really absorb right now. I did however find what looks to be a promising text just today. Andrzej Grzegorczyk's Outline of mathematical logic. So far I am very pleased. $\endgroup$– semwilSep 25, 2015 at 1:30
-
$\begingroup$ For me, an authors style goes a long ways in understanding what the author is trying to communicate before I attempt to convince myself if I agree. $\endgroup$– semwilSep 25, 2015 at 1:32