Uusi opinto-opas (sisältäen myös opetusohjelmat) lukuvuodelle 2018-2019 sijaitsee osoitteessa https://opas.peppi.utu.fi . Tältä sivustolta löytyvät enää vanhat opinto-oppaat ja opetusohjelmat.

The new study guide (incl. teaching schedules) for academic year 2018-2019 can be found at https://studyguide.utu.fi. This site contains only previous years' guides.

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Arkistoitu opetusohjelma 2015–2016
Selaat vanhentunutta opetusohjelmaa. Voimassa olevan opetusohjelman löydät täältä.
MATE6013 Logic, mathematics, computation and cognition 5–10 ECTS
Period I Period II Period III Period IV
Language of instruction
Type or level of studies
Advanced studies
Matematiikan tutkinto-ohjelma (FM)
Department of Mathematics and Statistics

General description

The course targets three objectives (i) understand the "inner mechanics" of the three fundamental model theoretic theorems of logic, (ii) understand in a second round, the theorems from a computational point of view and (iii) draw consequences about what branch of mathematics logic is and that it is such a branch.

The three theorems are the completeness theorem of first order logic, the compactness theorem and the Lowenheim-Skolem theorem. First time round we prove them by "model theory" andu  linger on how models are constructed (from the very deductive system). Having isolated the trick(s) that make the theorem work, we give two "logics" where they fail: with the generalized quantifier "finitely many" and with "recursve models" only. We connect the (un-)provability of the theorems to the expressive power of the language/model theory.

In a second round, we return to these theorems from a computational point of view looking for recursive enumerations of valid sentences in contrast to provably sentences. we linger on the notion of computability, provability and their "cognitive reality", in terms of the limits of human computations including the connection between (i) proof and (ii) computation (iii) availability of a certain kind of strategy in a related "game".


Joseph Almog
Valtteri Arstila
Vesa Halava


16-Sep-2015 –
Wed 16-Sep-2015 - 16-Dec-2015 weekly at 12-14, M2, Quantum

Study materials

A textbook for the class will be provided + hand made lecture notes that unwind the mechanics of certain cunning proofs. We will give home work problems on the underlying notions of informal sets and proof in a deductive system.

Further information

The course is organized as a collaboration by Department of Behavioural Sciences and Philosophy and Depertment of Mathematics and Statistics. Thus the courses MATE6013 and FILY1411 are the same.