MATxxx

1. Propositional logic: Axioms and Rules of Inference. Boolean Algebras. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms. 2. Predicate Logic: Existential and universal quantification, free variables and substitutions. Discussion of the various axiomatic systems for first-order logic (including axioms and rules of inference). The power and the limitations of axiomatic systems for logic: Informal discussion of the completeness and incompleteness theorems. 3. Sets: Operations on sets. Correspondence between finitary set operations and propositional logic. Correspondence between infinitary operations and quantifiers. The power and limitations of the language of set theory: Informal discussion of the set-theoretic paradoxes and the need for axiomatic systems for set theory. 4. Relations: Properties of relations. Special relations: Equivalence relations, partially ordered sets, totally ordered sets. 5. Functions: Operations of functions, direct image and inverse image. 6. Well-ordered sets: Correspondence between well-ordering relations and induction. Correspondence between well-ordering relations and choice functions. 7. Introduction to computability. Classical models of computation (recursive functions, and Turing models). Limitations of computation (the Halting Problem, the busy beaver problem, fast-growing functions). Contemporary models of computation: Digital vs analog vs quantum computing.