Model Theory:

Model Theory by Anand Pillay (AMS Notices, 2000)

Introduction to Model Theory by David Marker (MSRI)

First-order Model Theory by Wilfrid Hodges (SEP, 2005)

Random Graphs

The strange logic of random graphs by Joel Spencer (Springer, 2001)

Random Graphs and the Parity Quantifier by Phokion Kolaitis and Swastik Kopparty (STOC 2009)

Infinitary logic and admissible sets

Barwise: Infinitary Logic and Admissible Sets by H. Jerome Keisler and Julia F. Knight (BSL, 2004)

Infinitary Logic by John L. Bell (SEP, 2006)

Applications to algebra and algebraic geometry

Model Theory, Algebra and Geometry by Deirdre Haskell, Anand Pillay, and Charles Steinhorn (MSRI/Cambridge, 2000)
[full text]

Nonstandard models of arithmetic

Tennenbaum's Theorem by Richard Kaye

Nonstandard analysis

Non-measurable sets via nonstandard analysis by Terry Tao (blog, 2008)

Nonstandard probability by S. Michael Webb (NDJFL, 1975)

Quantifier elimination algorithms

A Survey of Quantifier Elimination: Syntactic and Semantic Approaches by Ting Zhang

QEPCAD: Quantifier Elimination by Partial Cylindrical Algebraic Decomposition

Decidable rings and Hilbert's 10th Problem

Hilbert's Tenth Problem over rings of number-theoretic interest by Bjorn Poonen (AWS, 2003)

Undecidability in number theory by Bjorn Poonen (AMS Notices, 2008)

Computable Model Theory

Pure Computable Model Theory by Valentina Harizanov (Handbook of Recursive Mathematics, 1998)

(Other topics to appear throughout the course.)

Computability Theory:

Incompleteness and Undecidability:

Set Theory: