Mathematical logic
Description of the module:
In this resource, we will know
:Basic
principles of writing a mathematical proof; Set
theory; Propositional
calculus and predicate calculus; Good order
and proof by recurrence. Target audience:
Students of 2nd year bachelor maths
Essential prerequisites:
The basic vocabulary of Algebra1.
Objectives:
1/ Acquire the foundations of mathematical reasoning.
2/ Acquire the foundations of set theory.
3/ Acquire the elements of writing mathematical proofs.
Content : Chapter
1: Introduction Elements of
mathematical language: axiom, lemma, theorem, conjecture. Writing
mathematical proofs: Basic principles of writing a mathematical proof. The
expression ‘without loss of generality’. Constructive proof and existential
proof. Chapter
2: Set theory Naive set
theory. Set definition of Cartesian product. Sets of parts. Assemblistic
definition of relations. Assemblistic definition of applications. Russel's
paradox. Other versions of Russel's paradox (Liar's paradox, Librarian's
paradox, Cretan liar's paradox). Optional: Zermelo-Fraenkel theory. Equipotence
relation. Cardinality of sets. Cantor-Betnestein theorem. Countable set, power
of continuum. Continuum hypothesis. Paul Cohen's theorem. Axiom of choice.
Godel's theorem. Chapter
3: Propositional
calculus and predicate calculus The logical
proposition, conjunction, disjunction, implication, equivalence, negation. The
truth table. The logical formula, tautology, contradiction. Rules of
inference or deduction, Modus Ponens rule. Modus Tollens rule. Predicate
calculus, Universal and existential quantifiers, The single existence
quantifier. Multiple quantifiers, Negation of a quantifier, Quantifiers and
connectors. Chapter
4: Good order and proof
by recurrence Recall
proof by recurrence. Theorem of proof by recurrence. Proof by
strong recurrence. Example of the existence of a prime decomposition of a
natural number. Optional (Proof by Cauchy recurrence. Proof of Cauchy Scwhartz
inequality by recurrence). Well-founded
order. Proof by the good order principle. Zermelo's general good order theorem.