(in Polish) Niestandardowe modele arytmetyki 1000-1M22NMA
Peano Arithmetic (PA) is the canonical theory axiomatizing the properties of the set of natural numbers with addition and multiplication. Via a standard translation between the languages of arithmetic and set theory, PA can be treated as the "theory of finite mathematical objects, i.e. Zermelo-Fraenkel set theory with the axiom of infinity replaced by its negation.
It follows from the basic theorems of mathematical logic that PA has models that are nonstandard, in the sense of not being isomorphic to the intended model. The course will be an introduction into the topic of nonstandard models of arithmetic. We will discuss not only classical results concerning the structure of nonstandard models, but also applications (sometimes quite recent) of nonstandard models in the proofs of theorems on the unprovability of certain statements in particular axiom systems.
During the semester we will cover the following topics:
1. PA and its fragments. Connections between definability and computability. Basic facts about the structure of models: order-type, cuts, end-extensions.
2. Types in arithmetic. Recursively saturated models, resplendence. Pointwise definable models, separations between fragments of PA.
3. Weak König's Lemma (WKL). Constructing models by means of the arithmetized completeness theorem. Scott sets and standard systems.
4. Advanced results about cuts and end-extensions. The Friedman and Tanaka self-embedding theorems. The MacDowell-Specker theorem on elementary end-extensions.
5. Proving unprovability by means of cuts: semiregular cuts, the so-called indicator method, partial conservativity of WKL over primitive recursive arithmetic.
6. Modern results: the Patey-Yokoyama theorem on partial conservativity of Ramsey's theorem for pairs over over primitive recursive arithmetic. The problem of characterizing the arithmetical consequences of Ramsey's theorem for pairs.
Depending on time and the interests of participants, we may also cover or mention additional topics, such as the Paris-Harrington theorem, automorphisms of models of arithmetic, cardinal-like models.
Type of course
Mode
Prerequisites (description)
Learning outcomes
The student:
1. knows the definition of Peano Arithmetic (PA) and understands its role in the foundations of mathematics.
2. understands the notion of a nonstandard model of arithmetic and is familiar with basic facts about the structure of such models.
3. is familiar with classical theorems concerning extensions and substructures of nonstandard models.
4. knows examples of applications of nonstandard models to separating axiomatic theories and proving unprovability results.
Assessment criteria
Exam.
Bibliography
1. Richard Kaye, Models of Peano Arithmetic, Oxford 1991.
2. Roman Kossak, James H. Schmerl, The Structure of Models of Peano Arithmetic, Oxford 2006.
3. Tin Lok Wong's course notes on "Model theory of arithmetic", https://blog.nus.edu.sg/matwong/teach/modelarith/
4. Selected research papers.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: