# PRINCIPLES OF MATHEMATICAL LOGIC

## Learning outcomes of the course unit

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In-depth study of logical concepts already seen in previous courses, with particular reference to the concept of "definition". An overview of other Mathematical Logic topics

## Prerequisites

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A course in basic Mathematical Logic

## Course contents summary

The programme will be agreed with the students, with a choice being made from the following topics:

1) Theories (Extensions and equivalent theories; conservative, defining and linguistic extensions. Theories with classes). Syntactical interpretations and syntactic models among theories. Conservative interpretation of ZF in GB.

2) The problem of foundations (Frege's Logicism and antinomies. Russell's types. Intuitionism. Opportunity of axiomatizing sets).

3) Universal algebra (algebraic structures, sets of generators, free algebras. Direct products, congruencies and quotients. Equational classes, Birkhoff's theorem).

4) Elements of theory of Categories (Yoneda's lemma. A set foundation: MMU. Added functors. Universal problems).

5) Algebraic and categorial logic.

6) Higher-order languages and multiset languages.

## Recommended readings

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S. Burris, H.P. Sankappanavar, A course in Universal Algebra, SPRINGER 1981;

W.S. Hatcher, Fondamenti della Matematica, BORINGHIERI 1973;

S. Mac Lane, Categorie nella pratica matematica, BORINGHIERI 1977;

M. Makkai, G. Reyes, First order categorical logic. Model-theoretical methods in the the ory of topoi and related categories, Lecture Notes in Mathematics, Vol. 611, SPRINGER-VERLAG, BERLIN-NEW YORK, 1977;

B. Mitchell, Theory of Categories, Academic Press 1965;

B. Mitchell, Introduction Category Theory and Homological Algebra, III ciclo C.I.M.E. 1971, EDIZIONI CREMONESE 1973;

H. Rasiowa, R. Sikorski: The Mathematics of Metamathematics, Warsaw 1970;

M. Servi, L'ABC delle categorie in cinque lezioni, Rapporto Matematico n. 258, Dipartimento di Matematica dell'Universita di Siena, 1993.

## Teaching methods

Another professor collaborates on the course. Theoretical lectures with assignment of exercises, which will then be corrected in class by the other students (in turns).