- Boolos, Burgess, and Jeffrey,
__Computability and Logic__ - Enderton,
__A Mathematical Introduction to Logic__

- *Chang and Keisler,
__Model Theory__ - Chapters 1-3 (1970s editions).
- Wilfrid Hodges,
__A Shorter Model Theory__ - Chapters 1, 2, 3.1-2, 4.1-3; 5.1, 5.2, 5.6, 6.2-3, 7.3.
- Marker,
__Model Theory: An Introduction__ - This book contains some very useful information on quantifier elimination.
- Poizat,
__A Course in Model Theory__ - Chapters 1-5 cover basic model theory topics like compactness and Lowenheim-Skolem, Chapter 6 contains examples and some quantifier-elimination proofs, and Chapters 9-10 cover the more advanced model theory prelim topics, like saturation, countable categoricity, and omitting types.
- Justin Bledin,
__An Even Shorter Model Theory (Fall 2006)__ - Justin posted some of his notes from 225A on his website. He says that edits and comments are welcome!
- Shelah,
__Classification Theory__ *Just kidding. The only reason to open it before your prelim is to make the other books you're working from seem much kinder and gentler afterwards.*

- *Robert Soare,
__Recursively Enumerable Sets and Degrees__ - Chapters 1-4 are all you need, but it would be a good idea to read the parts of Chapter 5 that go through the construction of a simple set.
*He's working on a new edition, and there is a preliminary version of the first few new chapters in circulation at Berkeley. It seems that some material has been added to the first four new chapters that has not historically been on the prelim (such as games).*- Cutland,
__Computability: An Introduction to Recursive Function Theory__

- *Richard Kaye,
__Models of Peano Arithmetic__ - The whole book is useful, but take a careful look at the technique for encoding subsets of N using nonstandard elements as products of primes (Chapter 1) and its discussion of overspill.
- *Barwise (ed.),
__Handbook of Mathematical Logic__, D.1., "The Incompleteness Theorems" by Smorynski (pp. 821-865) *This article provides a metamathematical overview of the incompleteness theorems.*- Judah and Goldstern,
__The Incompleteness Phenomenon__ *This book provides technical proofs of the incompleteness theorems.*- Smullyan,
__Godel's Incompleteness Theorems__ *This book provides technical proofs of the incompleteness theorems.*

- Tarski, Mostowski, and Robinson,
__Undecidable Theories__ - Chapters to follow.
- Schoenfield,
__Mathematical Logic__ - Chapter 6. Pay particular attention to the exercises!
- Enderton,
__A Mathematical Introduction to Logic__ - There is a short section that discusses whether certain reducts of PA are decidable, finitely axiomatizable, definability, etc. (p. 250, 1972 edition).

- Kunen,
__Set Theory__ - Chapter 1.
- Levy,
__Basic Set Theory__ *How much?*