Math 557: Mathematical Logic

Fall 2005, 3 credits, Mon-Wed-Fri 10:10-11:00, 101 Pond, Schedule 501343

Stephen G. Simpson, 333 McAllister, 863-0775, simpson@math.psu.edu

This course is suitable for all mathematics graduate students.  The
textbook will consist of notes provided by the instructor.  A version
of the notes is on line at
http://www.math.psu.edu/simpson/courses/math557/.

1. The Propositional Calculus  
  
Boolean operations, truth assignments, the tableau method, the
Completeness Theorem, the Compactness Theorem, combinatorial
applications.

2. The Predicate Calculus
  
Quantifiers, structures, satisfiability, tableaux, the G"odel
Completeness Theorem, the Compactness Theorem.

3. Proof Systems for Propositional and Predicate Calculus
  
Hilbert-style systems, Gentzen-style systems, the Interpolation
Theorem.

4. Extensions of the Predicate Calculus
  
Predicate calculus with identity, predicate calculus with operations,
categoricity, countable categoricity, many-sorted predicate calculus.
  
5. Theories, Definability, Interpretability
  
Mathematical theories (groups, fields, vector spaces, graphs, ordered
structures, ...), foundational theories (arithmetic, geometry, set
theory, ...), practical completeness, definability, implicit
definability, Beth's Theorem, interpretability.

6. Arithmetization and Incompleteness
  
Primitive recursive functions, representability, G"odel numbering, the
Diagonal Lemma, Tarski's Theorem on Undefinability of Arithmetical
Truth, G"odel's Incompleteness Theorem, Rosser's Incompleteness
Theorem, G"odel's Theorem on Unprovability of Consistency.