Model theory of arithmetic

The aim of this course is to introduce to the students the model theory of (natural number) arithmetic, and its connections to other branches of logic.

The lectures took place in the Winter Semester of 2014/15 at the Kurt Gödel Research Center. Each of them was 95 minutes long, with a five-minute break in the middle.


  • The Completeness Theorem and the Compactness Theorem for first-order logic, including their proofs.
  • The notions of recursive and recursively enumerable sets (aka computable and computably enumerable sets) of natural numbers.
  • Universal Turing machines (or one of their equivalents), the Halting Problem.
  • The coding of pairs, finite sequences, formulas, proofs, etc. in the natural numbers.
  • Gödel’s Incompleteness Theorems.
  • Definability of subsets in a structure.
  • Universal formulas are preserved in substructures, and existential formulas are preserved in extensions.
  • Groups, rings, …


Follow the links below for the notes in pdf. Please do let me know if you spot any typos or errors anywhere.

  1. The arithmetic hierarchy, 8 October 2014.

    Notes first uploaded on 11 October 2014; reference added and further exercises clarified on 18 October 2014

  2. Collection axioms, 15 October 2014.

    Notes first uploaded on 18 October 2014; typo in PA-minus corrected on 26 October 2014; Fact 2.4(b) corrected on 31 March 2017

  3. The Weak König Lemma, 22 October 2014.

    Notes first uploaded on 26 October 2014; Lemma 3.7 strengthened and figure added on 2 November 2014; typo corrected in the bibliography on 16 November 2014; proof of (b)⇒(a) for Proposition 3.3 fixed on 7 June 2022 (thanks to Jun Le Goh for pointing out the gap)

  4. The Arithmetized Completeness Theorem, 29 October 2014.

    Notes first uploaded on 2 November 2014; gap filled in using Std(L) on 9 November 2014

  5. Semiregular cuts, 5 November 2014.

    Notes first uploaded on 9 November 2014; comment added about the proof of Theorem 5.6, and reference removed on 16 November 2014; typo corrected in the definition of Γ‑PHP on 13 February 2015; Lemma 5.3 corrected (thanks to Paul J. Voda for pointing out the gap), the inequalities in Lemma 5.4(b) changed to strict ones, and the definition of Y in Further exercise (a) modified on 22 May 2017

  6. End and cofinal extensions, 12 November 2014.

    Notes first uploaded on 16 November 2014; exp added to Further exercise (d) on 17 November 2014; asterisk added to Theorem 6.8, and various references added or modified on 14 January 2017; two references to Kaye’s paper added, parameters added to Further exercise (b), and several mistakes in Further exercise (d) corrected on 6 February 2017

  7. Recursive saturation, 19 November 2014.

    Notes first uploaded on 22 November 2014; comment about uncountable recursively saturated structures added on 29 November 2014; typo in Theorem 7.6 corrected on 26 January 2015

  8. Back-and-forth, 26 November 2014.

    Notes first uploaded on 29 November 2014; various inaccuracies in Remark 8.7 corrected on 22 March 2017

  9. The Mac Dowell–Specker Theorem, 3 December 2014.

    Notes first uploaded on 7 December 2014; typos in the proof of Theorem 9.10 corrected on 9 December 2014

  10. Strongly definable types, 10 December 2014.

    Notes first uploaded on 14 December 2014; meaning of extensions for types clarified, and comment about P- and Q-points added on 1 January 2015; the inequality in the definition of ∃ changed to a non-strict one on 11 January 2015; several typos fixed in the proof of Proposition 10.4 on 7 June 2022 (thanks to Mengzhou Sun for reporting them)

  11. Minimal types, 17 December 2014.

    Notes first uploaded on 22 December 2014; labels added to Figure 11.3, and subsection on rarity and minimality expanded on 1 January 2015; modified in view of the new definition of ∃ on 11 January 2015; typos in the proof of Proposition 11.7 fixed, and the paragraph immediately after Further exercise (f) revised on 7 June 2022 (thanks to Mengzhou Sun for reporting the mistakes)

  12. Preservation, 7 January 2015.

    Notes first uploaded on 11 January 2015; typo in clause (ii) of the definition of isolated types fixed (thanks to Mengzhou Sun for reporting this), and the reference to the Dimitracopoulos paper corrected in the Further comments on 7 June 2022

  13. Arithmetic with a top, 14 January 2015.

    Notes first uploaded on 18 January 2015; various parts of the Further exercises corrected on 26 January 2015

  14. Satisfaction classes, 21 January 2015.

    Notes first uploaded on 25 January 2015; two figures added on 26 January 2015; Tarski’s clause (T1) clarified, corrections made in the proof of Theorem 14.3 in clause (U1), the defining property of rank, and item (1), Remark 14.4 changed to a comment and so subsequent theorem numbers are shifted accordingly, reference for Theorem 14.5 added, and various other minor changes and additions made on 14 August 2017

  15. Forcing in arithmetic, 28 January 2015.

    Notes first uploaded on 2 February 2015; the displayed condition in Proposition 15.7 strengthened, references added, and some extra explanations inserted on 24 June 2017; item [15] in the list of references updated, the unnecessary use of BΣ1 in the Kleene Normal Form Theorem eliminated, Corollary 15.6, Proposition 15.7, and Further Exercise (a) removed, and the numbering shifted accordingly on 16 April 2019

In the lecture notes, I list some suggestions for ‘Further reading’. Students are welcome, but not required, to follow all the pointers there.