# Master's program in mathematics – core modules and elective courses in Mathematical Logic and Theoretical Computer Science

This page contains information for (prospective) students of the mathematics master's program with specialization (*Schwerpunkt*) on *Mathematische Logik und Theoretische Informatik*.
Note that nearly all of the information listed here (admission, curriculum etc) is **not** in our responsibility. This information is provided for your convenience, but we cannot take any responsibility that it is correct, complete or still up to date. Please make sure to check with the authoritative sources, e.g.

## General information for foreign students

This paragraph is relevant mainly for foreign students who are interested to come to Vienna to specialize in mathematical logic and theoretical computer science. (Austrian students will know all the information provided here anyway.)

- Admission
- The KGRC is not involved in any way in questions of enrolement, admission, tuition fees etc. Please consult StudentPoint or the SSC Mathematik or help.gv.at. In particular, the following points is not in the responsibility of the KGRC and provided for information only. Also, we cannot answer questions regarding the following points, please contact StudentPoint instead.
- Tuition fees
- The tuition fees are relatively low (363.36 to 726.72 euros per semester). EU citizens might have this fee waived in certain situations.
- Cost of living
- Master students do not receive salaries, and there are hardly any fellowships or grants available for them. The parents are supposed to cover the living expenses and tuition fee of the student. Only in exceptional cases, for EU citizens, when the income of the student and the student's parents is very low, and the student's progress is very good (which is measured in a rather bureaucratic and inflexible way), then a goverment grant is provided (up to 600 EUR per month, if the student's family does not live close to Vienna).
- Duration
- It is probably quite hard to finish a study program in the official timeframe of 4 semesters, unless you already have a good education in mathematical logic to begin with. Note that it is not a problem to take longer than the 4 semesters, but of course you will then have to provide the cost of living for a longer period of time (and in the rare case that you get the goverment grant mentioned above, this grant will be only paid for up to 5 semesters).
- Language
- Basic German skills (or at least to take a German class once you arrive) are recommended, but probably not necessary: While admission to the master's program in principle requires a proof of proficiency in German, this can be (and usually is) waived during the admission process. And it is probably possible to finish the study program by attending classes in English only: All the lectures at the KGRC are held or can be held in English. You will also have to take courses outside th KGRC (at least for the
*Verbreiterung*(courses from other areas of specialization; see below), but possibly also for*Wahlfach*(electives) or for logic courses at TU). Some of these courses are in English anyway, and many more can be held in English if requested (just contact the lecturer and ask). But of course we can not guarantee anything for non-KGRC courses.

## Courses of the specialization in mathematical logic and theoretical computer science

As you know from the curriculum, the following courses are required for the master's program with specialization in logic, adding up to 120 ECTS:

### Core modules in mathematical logic and theoretical computer science (Pflichtmodule: MLOL, MLOM, MLOI, MLOS), 30 ECTS

The mandatory courses are held at the KGRC every year. In the following table, *W* indicates that they are held in the winter semester, *S* in the summer semester and *S/W* in either semester.

Module | When | ECTS | SSt | Type | Name |
---|---|---|---|---|---|

MLOL | W | 6 | 4 | VO | Introduction to Mathematical Logic |

MLOL | W | 3 | 2 | PS | PS Introduction to Mathematical Logic |

MLOM | S | 5 | 3 | VO | Axiomatic Set Theory 1 |

MLOM | S | 3 | 2 | PS | PS Axiomatic Set Theory 1 |

MLOI | W | 5 | 3 | VO | Introduction to Theoretical Computer Science |

MLOS | S/W | 8 | 4 | SE | two seminars (SE), 4 ECTS and 2 SSt each |

### Electives from your chosen area of specialization, mathematical logic and theoretical computer science (Vertiefungs LVAs: MLOV), 21 ECTS

For this part you can use every course with the code MLOV. At the KGRC we regularly give the following courses, but not all of them every year: Model Theory and Computability and Complexity alternate in summer semesters. The Research Seminar is held every semester (it is an advanced seminar suitable for PhD students and advanced master students). Note that there are many courses at TU that can be credited (*angerechnet*) as MLOV, see the section on TU.

Module | When | ECTS | SSt | Type | Name |
---|---|---|---|---|---|

MLOV | (S) | 3 | 2 | VO | Model theory |

MLOV | (S) | 3 | 2 | VO | Computability and Complexity |

MLOV | W | 7 | 2 | VO | Set Theory (reading course) |

MLOV | W | 4 | 2 | VO | Selected Topics in Mathematical Logic |

MLOV | S | 4 | 2 | VO | Selected Topics in Set Theory |

MLOV | W & S | 4 | 2 | SE | Research Seminar (Mathematical Logic) |

### Courses from other areas of specialization (Verbreiterung), 24 ECTS

You need courses from other areas of specialization in mathematics master programs, i.e. courses that have a code **not** starting in MLO. Note that you do not have to choose all these courses from one single non-logic area of specialization; you can use courses from more than one. Of these 24 ECTS, at least 15 ECTS have to be from the *Standardausbildung*, and at most 4 ECTS from seminars.

### Further electives (Wahlfach), 15 ECTS

Any course from the Mathematics master program will do, including courses at the KGRC that you do not use for core modules or electives in mathematical logic and theoretical computer science. Note that you can also use TU courses, see below.

### Thesis (Masterarbeit), 30 ECTS including defense

A master thesis can be supervised by faculty with habilitation (at the KGRC currently S. Friedman, B. Miller and V. Fischer). In practice we can offer topics from set theory and to some degree model theory and recursion theory. If you would prefer a thesis in a research area that is represented at TU (such as proof theory or automatic theorem proving), it will be necessary to get a supervisor from TU. It is theoretically possible to have an external advisor, but this is cumbersome and uncommon for master theses (but you have to discuss it with the SPL Mathematics). As alternatives, the thesis can still be supervised at the KGRC if you can get a secondary supervisor from TU.

## Recommended time plan

In the following, let us assume that your aim is to write a master thesis at the KGRC. It is required that you already know some very basic logic as covered, e.g., by the Bachelor course *Grundzüge der Mathematischen Logik* (Basic Concepts of Mathematical Logic). Ideally, you already took *Grundbegriffe der mathematischen Logik* (Basic Concepts of Mathematical Logic) and *Axiomaische Mengenlehre 1* (Axiomatic Set Theory 1) while studying in the bachelor's program. In this case, a possible schedule for the remaining Logic courses would be as follows:

TO DO: GIVE EXAMPLE PATH THROUGH MASTER'S PROGRAM

Notes:

- Don't forget that you additionally need 24 ECTS worth of
*Verbreiterung*courses, i.e., non-logic courses. - You can take the
*reading course*twice (as suggested in the time plan above), or even three times, provided of course that the contents is different (and suitable) each time. So the first time you will learn, e.g., about forcing and measurable cardinals; the next course about descriptive set theory and more advanced forcing techniques, etc. The same applies to the*Ausgewählte Kapitel*(Selected Topics) lectures. - If you did not do the
*Logik*and*Mengenlehre*during the Bachelor program, you will obviously have to compress the master a bit. - Of course many variations are possible. E.g. you can take many additional courses at TU. You definitely should do that if you are interested in the computational side of logic.
- Or if you do not want to write a master thesis in set theory, then the set theory reading courses as well as the
*Ausgewählte Kapitel aus Mengenlehre*(Selected Topics (Set Theory)) should be subsituted by courses from the area of the thesis.

## Mathematical Logic at the University of Technology (TU Wien)

At the Technische Universität Wien (TU for short), there are several active research groups and many interesting courses on mathematical logic and theoretical computer science.
You are encouraged to enroll at TU in addition to the *Universität Wien* (univie for short).

### How to enroll at TU and what TU curriculum to choose

- For general information, contact the TU Studienabteilung.
- You can enroll in any additional curriculum at TU you want, or just choose
*Mitbelegung*(co-registration): - The simplest version is
*Mitbelegung*: You do not enroll in a TU program, but rather just register your univie curriculum at TU. A page at the TU Studienabteilung has more information. - Alternatively, you can enroll for any curriculum offered at TU. For this, you need to provide the usual documents:
*Reifeprüfung*for a Bachelor program and a Bachelor Diploma for a Master Program, which might be complicated for foreigners. Again, you can find more information (in German) at the Studienabteilung. Natural choices for curricula would be:

Master program Computational Logic

Master program Technische Mathematik

Master program Computer Science - Usually, additional enrollment at TU should not incur any fees nor should it interfere with any potential financial support you might be eligible for, such as
*Familienbeihilfe*(family allowance); but please check this yourself!

### When you should enroll

- We recommend that you enroll as soon as possible (and even in semesters in which you do not plan to attend any courses at TU).
- The reason: To be able to take an exam for a course at TU, you have to be enrolled at TU both in the semester of the course as well as in the semester of the exam. So if you decide near the end of the master program that you would like to take an exam for a TU course that was given in a previous semester, it is necessary that you have been enrolled in that semester.

### How to get credit using TU courses for the univie master program

- Whether a TU course can be used is not decided by the KGRC, but rather by the SPL Mathematics. The SPL decides on an individual basis whether a course from another curriculum will be accepted (
*angerechnet*) for the univie master. - Theoretically, you can (or even should?) ask the SPL whether a TU course will be accepted before taking it (otherwise your effort might be wasted). However, the SPL is probably not very interested in such questions, and we never had any problems getting reasonable courses accepted. (Of course it has to be a specialized and reasonably advanced course on logic or theoretical computer science; an easy general introduction to logic will not be accepted as a MLOV course, since it would add redundancy to the Bachelor course
*Grundzüge*and the mandatory master courses.) - So while we (the KGRC) cannot guarantee anything, we would recommend to just take any reasonable course at TU and you can be pretty sure that it will be accepted by the SPL.

### Why you should enroll, what courses you should take

While the KGRC offers all mandatory courses for logic and theoretical computer science as an area of specialization in your master's program, as well as sufficiently many ECTS points for the electives in mathematical logic and theoretical computer science (MLOV) part, there might be good reasons to take some additional courses at TU:

- The KGRC offers enough ECTS to cover the core modules and electives part of the master's program, but there is not very much left for the further electives part. So while you are certainly welcome to take courses from other parts of mathematics, it might be less effort and/or more interesting for you to take logic courses at TU. (In particular, since you will have to write a master thesis, you will have to learn a lot about the area(s) that interest(s) you anyway. For example if you write a thesis in set theory, Martin Goldstern's courses will be interesting for you.)
- The KGRC specializes on the
*semantic*part of logic (set theory, model theory, a bit of recursion theory etc). There is a different side to logic as well, leaning more towards syntax and computer science, involving subjects such as nonclassical logics, proof theory, automatic theorem proving, model checking. All these subjects are represented in research and teaching at the TU. In particular, we recommend to take*Beweistheorie*(proof theory) at TU, one of the four classical branches of mathematical logic, which is currently not represented at the KGRC. (But even if you are only interested in set theory, you should check out Martin Goldstern's courses at TU, as already mentioned.)

### TU Research groups in (or close to) mathematical logic

(The following list is probably not complete)

- Algebra Group, led by Martin Goldstern, emphasis on set theory and universal algebra.
- Computational Logic Group, led by Matthias Baaz, emphasis on proof theory and non-classical logics.
- Theory and Logic Group Group, led by Alexander Leitsch, emphasis on proof theory and theoretical computer science.
- Database and Artificial Intelligence Group, led by Georg Gottlob and Reinhard Pichler
- Knowledge-Based Systems Group, led by Thomas Eiter
- Formal Methods in Systems Engineering, led by Laura Kovács, emphasis on Model Checking