Foundations of mathematics

Course goals

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Content

The course Foundations of Mathematics is an elective course for mathematics students. The course is prerequisite knowledge for students who want to further specialize in logic. For more information about study tracks, see the student website.

Learning goals: 
The topics covered in this course are:

• Elementary (naive) set theory: calculating with cardinalities, Axiom of Choice, Zorn's Lemma, Well-orderings, Transfinite Induction and Recursion, Equivalence of Axiom of Choice, Zorn's Lemma and Well-ordering Theorem.
• First-order languages, formulas, sentences, theories.
• Structures and the concept "sentence - is true in structure M".
• Definability and the Compactness Theorem.
• Some model theory: elementary substructures, quantifier elimination, Löwenheim-Skolem theorems, categorical theories, Łos-Vaught Test.
• Proof trees.
• Completeness Theorem.
• The axiom system of Zermelo-Fraenkel.
• Ordinal and cardinal numbers; the cumulative hierarchy.
• The real numbers in ZF.

After completing the course, the student will be able to:

• perform simple cardinality calculations;
• produce proofs using the Axiom of Choice and Zorn's Lemma;
• define functions and sets using Transfinite Recursion;
• write sentences in a formal first-order language of a given structure to determine whether they are a model of a given theory;
• apply the Compactness Theorem;
• apply quantifier elimination and the Löwenheim-Skolem theorems;
• create formal proof trees for simple tautologies;
• write simple mathematical statements in the formal language of ZF.

The student will be able to independently select the relevant proof techniques for a given simple problem and solve the problem with them, and clearly articulate the solution to the problem.

Course page: 
https://leoguetta.github.io/grondslagen2025.html

Course format: 
There are lectures twice a week for two hours each, and tutorials twice a week for two hours each.

Assessment: 
There is a final exam and a resit covering all material. Additionally, there are 7 hand-in assignments that count for 15%: if T is the exam grade (or the resit grade) and I is the grade for the assignments, then the final grade is the maximum of T and (85T+15I)/100. The assignments thus remain valid through the resit.

Retake and effort requirement: 
Students who have a final grade lower than 4 may only participate in the resit if they meet the course effort requirement, namely: at least 5 assignments completed.

Course language: 
The course is taught in English.