I am looking for a good book about model theory. As this is obviously too vague, let me
explain what I am looking for and why.
First I am interested about the basics and foundations of model theory. Right now I am not interested in their applications (like proving things in mathematics or even independence results like Cohen's — but of course it is not a problem if a book deals with some of these applications if it does not only that).
Second, until a few days ago I believed I knew well enough what was a model.
But since two days I am not so sure. I have a problem with the notion of model inside ZF (or ZFC, or any formalized set theory) of a theory, and in particular, with the meaning of the satisfaction relation in this case. I would like a book which treats that aspect.
These problems arose with my trying to understand the answers to my question
A meta-mathematical question related to Hilbert tenth problem
I am currently having endless discussions in comments about that notion what I am said doesn't make sense to me (and the converse is clearly also true). A good book will certainly help me and save time for my respectable interlocutors. Thanks…
Best Answer
Basic model theory texts are Marker's Model Theory; An Introduction and A Shorter model theory by Hodges. Maybe the one on Mathematical Logic by Cori and Lascar too. I'm not sure you need a book which specifically treats this aspect but a general understanding of what a theory, and a model of a theory (e.g. ZF or ZFC) is should do (the first chapter of Marker's book covers this). Once you know the basics then just think of a language with a binary relation symbol (for set membership) and formulas (in the language) for the axioms of ZF. Then you can think of a model of this etc...