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mathcourse - 2009S_Foundations_of_Mathematics.wiki

Course Information

Final Exam

Thursday, April 23, 10:00 a.m. - 11:50 a.m.

Prerequisites

This course is an introductory course on the foundations of mathematics from the point of view of higher order logic. No previous courses in set theory, logic, or computers are required. Mathematical maturity is strongly recommended, as evidenced by successful completion of other upper-level undergraduate mathematics courses.

Content

A century ago, set theory was used to establish solid foundations for mathematics. More recently, solid foundations have been established with higher order logic. This course will cover

  • The syntax of mathematical expressions
  • Parsing and printing mathematics in a precise form
  • The rules of higher order logic
  • Type theory for higher order logic
  • Axioms at the foundation of mathematics
  • Proving the rules of logic
  • A proof of the law of induction
  • Recursion and recursive types
  • Sets
  • The construction of the set of integers and the set of real numbers
  • The structure of mathematical proofs

Objectives

Students who take this course will develop an understanding of the foundations of logic and mathematics. This will give a deeper understanding of all other mathematics courses, including geometry, applied math, and algebra. It will give a unified perspective from which all other math subjects can be understood.

Lectures

The course will be based on lecture notes written by Hales.

Evaluation

Grades will be based on quizzes, homework, and exams.

  • 20%. quiz every two weeks.
  • 30%. weekly problem set.
  • 20%. one midterm exam.
  • 30%. final exam.

Some materials on HOL

These materials are not textbooks for an undergraduate math course. They are primarily intended for people who implement the HOL system on a computer. Many parts, therefore, will not be relevant to the course. However, you may find certain parts useful.

Some Books of Related Interest