2025
Mathematical Logic
Name: Mathematical Logic
Code: MAT14355M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Presentation
Mathematical Logic has strong links with the foundations of Mathematics and Computer Science. In this UC we explore the applications of formal logic in mathematics including the study of the expressive power of formal systems and the deductive power of formal mathematical proof systems.
Sustainable Development Goals
Learning Goals
- Capacity of formalizing in terms of well-formed formulae..
- Insight in the fundamental properties of propositional and first-order logic.
- Capacity to relate problems of consistency of collections of formulae to the existence of models, i.e. formalized structures in an axiomatic set theory.
- Insight in the diversity of mathematical structures associated to an axiomatics and the axiomatic theories valid in a given mathematical structure.
- Insight in the fundamental properties of propositional and first-order logic.
- Capacity to relate problems of consistency of collections of formulae to the existence of models, i.e. formalized structures in an axiomatic set theory.
- Insight in the diversity of mathematical structures associated to an axiomatics and the axiomatic theories valid in a given mathematical structure.
Contents
- Propositional logic. Syntax and Semantics. Deduction and logical implication. Metatheorems of Soundness and Completeness, consistency, compatibility. Fundamental properties of Propositional logic: Interpolation, compactness, decidibility.
- First-order logic. Syntax, deduction. Semantics, models, logical implication. Metatheorems of Soundness and Completeness, compactness and the Theorem of Löwenheim-Skolem. Applications: formal and informal reasoning, nonstandard models of arithmetic.
- First-order logic. Syntax, deduction. Semantics, models, logical implication. Metatheorems of Soundness and Completeness, compactness and the Theorem of Löwenheim-Skolem. Applications: formal and informal reasoning, nonstandard models of arithmetic.
Teaching Methods
Systematic introduction to logical systems and models.
Illustration of concepts and arguments of the theory by means of concrete examples.
Processing by exercises and applications in other areas of mathematics.
Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of solving individually the two lists of exercises in which themes or problems related to the syllabus are addressed. The final classification is the average of the classifications on the lists.
2) Evaluation by exam.
Formative evaluation is done in to improve the learning process; the elements of formative evaluation will have no weight on the final classification.
Illustration of concepts and arguments of the theory by means of concrete examples.
Processing by exercises and applications in other areas of mathematics.
Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of solving individually the two lists of exercises in which themes or problems related to the syllabus are addressed. The final classification is the average of the classifications on the lists.
2) Evaluation by exam.
Formative evaluation is done in to improve the learning process; the elements of formative evaluation will have no weight on the final classification.
Teaching Staff
- Bruno Miguel Antunes Dinis [responsible]
Certificações
