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Part III (MMath/MASt)

 

Please take this page in conjunction with the Part III Guide to Courses Foundations section.

Traditionally, mathematical logic is divided into areas such as category theory, computability theory, model theory, proof theory, and set theory. The Part III courses in Foundations usually cover the areas category theory, model theory, and set theory:

Category theory and model theory deal with general constructions and concepts that can be applied in many other areas of mathematics: as a consequence, these courses will use examples from courses such as Part Ia Groups, Part Ib Groups, Rings, & Modules, and Part Ib Metric & Topological Spaces. Students are expected to be familiar with the main definitions and techniques from these courses or willing to achieve that familiarity by reading the corresponding textbook literature.

Courses in set theory are given under various names, e.g., Topics in Set Theory or Infinite Games. Some parts of set theory closely link to the general theory of metric and topological spaces, so Part Ib Metric & Topological Spaces can be relevant.

All Part III Foundations courses build on the material of the Part II course Logic & Set Theory. This course covers the basics of mathematical logic (syntax and semantics of propositional and first order logic, completeness and compactness) as well as axiomatic set theory (axioms of Zermelo-Fraenkel set theory, ordinals, cardinals). The amount of knowledge in logic and set theory assumed in the Part III logic courses differs by course:

  • Logic: essential for courses in model theory and set theory; important for courses in category theory.
  • Set Theory: essential for courses in set theory; important for courses in model theory.

Logic

The Part III Foundations courses generally assume that students are familiar with syntax and semantics of first order logic, Gödel's Completeness Theorem, and its consequences, such as compactness and various results on non-definability or non-axiomatisability.

As a reality check, have a look at Examples 8 to 12 on Example Sheet #3 of the 2018/19 version of Part II Logic & Set Theory.

If you feel that you need to brush up on your logic background, the following resources will be useful:

  • The lecture notes Logic and Set Theory by Imre Leader. These notes formed the basis of the Part II course Logic & Set Theory as offered in the academic years 2017/18 and 2018/19.
  • Peter T. Johnstone, Notes on Logic and Set Theory, Cambridge Mathematical Textbooks (Cambridge University Press, 1987).
  • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas, Mathematical Logic, Undergraduate Texts in Mathematics (Springer-Verlag, 1994).

Set Theory

(Some of the) Part III Foundations courses assume that students are familiar with the basics of axiomatic set theory such as the axioms of Zermelo-Fraenkel set theory, the Axiom of Choice, and ordinals and cardinals.

As a reality check, have a look at Examples 9 and 10 on Example Sheet #2, Example 6 on Example Sheet #3, and Examples 7, 8, 10, and 11 on Example Sheet #4 of the 2018/19 version of Part II Logic & Set Theory. You can also find a quick self-test with fourteen simple multiple choice questions here. You should easily get most of these right without any effort or preparation. Once you have done them, you can test if you are right by highlighting the text behind each bullet point, which is white so hidden.

  • Basics: 1C 2D 3C 4C 5D 6D
  • Ordinals: 7B 8C 9A
  • Cardinals: 10A 11D 12D 13D 14B

If you feel that you need to brush up on your logic background, the following resources will be useful:

  • The lecture notes Logic and Set Theory by Imre Leader. These notes formed the basis of the Part II course Logic & Set Theory as offered in the academic years 2017/18 and 2018/19.
  • Peter T. Johnstone, Notes on Logic and Set Theory, Cambridge Mathematical Textbooks (Cambridge University Press, 1987).
  • Kenneth Kunen, Set Theory, Studies in Logic Vol. 34 (College Publications, 2011).