(in Polish) Niestandardowe modele arytmetyki

elective monographs

The student is expected to be familiar with basic notions and facts from set theory and mathematical logic, for example at the level of the Introduction to Mathematics and Mathematical Logic courses at MIMUW.

Classroom

Peano Arithmetic (PA) is the canonical theory axiomatizing the properties of natural numbers. Via a standard translation into the language of set theory, PA can be treated as the canonical theory of finite sets and finite mathematical objects.

The course will be an introduction into the topic of nonstandard models of PA and its subtheories, that is models nonisomorphic to the intended one. We will discuss both results concerning the structure of nonstandard models and the applications of such models in proofs of unprovability.

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.

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.

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.

Exam.