Kaisa_2012_3_photo by Veikko Somerpuro

Models of arithmetic: Note room and time changes

We study the rich theory of models of arithmetic. You pass the course by presenting a project or by taking an exam (your choice).

A model of arithmetic is a model of the Peano axioms (first order). There is a rich theory of already the countable models of PA, of which there are continuum many (besides the standard one of course!). We will study the recursion and model theoretic aspects of these structures, covering Scott sets and standard systems, the arithmetized completeness theorem, and finally Goodstein sequences and the Paris-Harrington theorem, which was the first concrete independence result (in the eyes of some).

ROOM MONDAY IS NOW B322, WEDNESDAY IS NOW C122. Tuesday's class is in C122.

11.12.2017 at 09:00 - 2.5.2018 at 23:59


Juliette Kennedy's picture

Juliette Kennedy

Published, 17.1.2018 at 8:52

Please note the following room and time changes:

MON 12-14, B322
TUE 12-14, C122
WED 14-16, C122

Juliette Kennedy's picture

Juliette Kennedy

Published, 16.1.2018 at 13:37

Dear Students, Would it be possible for us to change the time of Tuesday's class to 12-14? That way we get a better room.

I was able to move the lectures on Monday into the room C322 and excercises on Wednesday into the room C122.

Juliette Kennedy's picture

Juliette Kennedy

Published, 15.1.2018 at 22:40

There is no example class this week, January 17th.


Here is the course’s teaching schedule. Check the description for possible other schedules.

Mon 15.1.2018
12:15 - 14:00
Tue 16.1.2018
10:15 - 12:00
Mon 22.1.2018
12:15 - 14:00
Tue 23.1.2018
12:15 - 14:00
Mon 29.1.2018
12:15 - 14:00
Tue 30.1.2018
12:15 - 14:00
Mon 5.2.2018
12:15 - 14:00
Tue 6.2.2018
12:15 - 14:00
Mon 12.2.2018
12:15 - 14:00
Tue 13.2.2018
12:15 - 14:00
Mon 19.2.2018
12:15 - 14:00
Tue 20.2.2018
12:15 - 14:00
Mon 26.2.2018
12:15 - 14:00
Tue 27.2.2018
12:15 - 14:00
Mon 12.3.2018
12:15 - 14:00
Tue 13.3.2018
12:15 - 14:00
Mon 19.3.2018
12:15 - 14:00
Tue 20.3.2018
12:15 - 14:00
Mon 26.3.2018
12:15 - 14:00
Tue 27.3.2018
12:15 - 14:00
Mon 9.4.2018
12:15 - 14:00
Tue 10.4.2018
12:15 - 14:00
Mon 16.4.2018
12:15 - 14:00
Tue 17.4.2018
12:15 - 14:00
Mon 23.4.2018
12:15 - 14:00
Tue 24.4.2018
12:15 - 14:00
Mon 30.4.2018
12:15 - 14:00

Other teaching

17.01. - 28.02.2018 Wed 14.15-16.00
14.03. - 28.03.2018 Wed 14.15-16.00
11.04. - 02.05.2018 Wed 14.15-16.00
Juliette Kennedy
Teaching language: English


The material for the course are the "famous" notes of Craig Smorynski. We will also rely on Richard Kaye's text "Models of Arithmetic".


Homework for example class meeting 24.01

Prove the compactness theorem for first order logic. Prove the validity of "Vaught's Criterion" for elementarily of a submodel, i.e. M is an elementary submodel of N if it is a submodel and if any existential formula which is true in N can be witnessed by an element of M. Prove the consistency of the theory T^C_\omega as in notes. Time permitting we will go through some basics of computability theory.

Homework for example class 31.012

Prove theorem 1.6 (ii) from chapter 1 of our notes, a version of overspill.
Prove the compactness theorem using ultraprodcuts. See attached notes.
Write down a Turing machine for 1. multiplication 2. exponentiation.

Homework for example class 07.02

1. Prove that if you add names to the language of a model M of arithmetic (so the language of M is the language of PA), then take a model M' of the extended language, then M' is a conservative extension of M, i.e. every set which is definable (with parameters) in M' is already definable (with parameters) in M.
2. Prove that the Cantor pairing function has a unique inverse.
3. Compactness for f.o. logic
4. Every set coded in the "D_d" sense is coded in the "F_x" sense

Homework for example class 21.02

Prove theorems 1.20 and 1.21 from the notes, as indicated in class. Finish coding problem, that every set coded in the "D" sense, has a code in the "F" sense.

Homework for example class 28.02

Read theorem 2.7 and corollary 2.9 in Kaye's Models of Peano Arithmetic and present it in class. Read section 3.1, prove theorem 3.3. Start reading chapter 9 (Satisfaction) up to theorem 9.15.

Homework for March 21

Prove the existence of a recursive, non-primitive recursive function.

In E. Mendelson's text Introduction to Mathematical Logic, 3rd edition, this is problem 3.33 on page 146.


Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Model theory of arithmetic including arithmetized completeness and Scott's Theorem, saturated models

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Basic theory of models of arithmetic including the arithmetized completeness theorem; Scott sets

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later