Logic exercise sessions at the ENS

During the years 2015–2016, 2016–2017 and 2017–2018, I taught exercise sessions for the mathematical logic lecture of the Department of Mathematics and Applications of the École normale supérieure in Paris. This course is usually attended by 1^st and 2^nd year students of the ENS (corresponding to their 3^rd and 4^th year of higher education). The corresponding lecture was taught by Zoé Chatzidakis (her lecture notes are available here) during the year 2015–2016, and by Todor Tsankov during the two next years.

You will find on this page my exercise sheets for the year 2017–2018 (in French); those are the most complete. The exercises are strongly inspired by those by my predecessor Silvain Rideau-Kikuchi, themselves inspired by those by Pierre Simon, themselves... (No, this is not an infinite decreasing sequence.) Don't hesitate to reuse my exercises if you want to.

• The sheet no. 1, about equinumerosity, orderings, and ordinals, and its solutions.


• The sheet no. 2, about ordinals, cardinals, and the axiom of choice, and its solutions. Here is also a proof of the equivalence between the two different definitions of both ordinal sum and product seen during the lecture.


• The sheet no. 3, about cardinals and Ramsey theory.


• The sheet no. 4, about first-order logic, compacity and Ramsey theory.


• The sheet no. 5, about compacity, elementary embeddings, and quantifier elimination.


• The sheet no. 6, about quantifier elimination.


• The sheet no. 7, about ultrafilters, ultraproducts, and primitive recursive mappings.


• The sheet no. 8, about computability.


• The sheet no. 9, about decidability and arithmetic.


• The sheet no. 10, about undecidability, incompleteness, and arithmetic.


• The sheet no. 11, about interpretability and set theory.


• The sheet no. 12, about consistency proofs in set theory.


• The sheet no. 13, about Gödel's second incompleteness theorem, the perfect set theorems, and Cantor-Bendixson analysis.