"The graduate program in mathematical logic at the University of Illinois at Urbana-Champaign is one of the most active such programs in the world, with a large community of faculty and students. Most aspects of the subject are represented here, especially model theory and its applications and descriptive set theory. A particular emphasis of our program is on the connections of logic to other important areas of mathematics."  from Logic, Department of Mathematics, University of Illinois at Urbana-Champaign

Books on logic can be found in the Mathemathics Library Stacks on the main level shelved under call number 511.3 to 511.4.

The discipline of Mathematical Logic is a rather historical umbrella encompassing the fields of Model Theory, Set Theory, Descriptive Set Theory, Computability Theory (also known as Recursion Theory), Proof Theory, and Universal Algebra. The research of the logic group at Illinois focuses on Model Theory (Lou van den Dries, Philipp Hieronymi) and Descriptive Set Theory (Anush Tserunyan).

Mathematical Logic (basic Model Theory and Computability Theory)

• L. van den Dries, "Mathematical Logic", lecture notes
• A. Tserunyan, "Mathematical Logic", lecture notes and First Order Logic and Gödel Incompleteness problem set

Model Theory

• K. Tent and M. Ziegler, "A Course in Model Theory"
• D. Marker, "Model Theory: An Introduction"

Descriptive Set Theory

• A. Kechris, "Classical Descriptive Set Theory"
• A. Tserunyan, "Introduction to Descriptive Set Theory", lecture notes and Descriptive Set Theory problem set

Set Theory

• J.-L. Krivine, Théorie des ensembles (French) 2ème édition (2007)
• K. Kunen, Set Theory (2011)
• A. Tserunyan, "Basic Set Theory", introductory lecture notes for undergraduates

Computability Theory

• L. van den Dries, "Recursion Theory Notes", lecture notes