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Philo. 106b Math Logic Professor Alan Berger e-mail: [email protected] Office Hours: TBA (On Wednesday late afternoon and/or and by appt.) Attention: If you are a student with a documented disability on record at Brandeis University and wish to have a reasonable accommodation made for you in this class, please see me immediately. Any case of dishonesty is a serious academic infraction and is subject to disciplinary action. This includes cheating on test, using other materials (includes Internet) without citing the source. Philosophy 106b (designed primarily for students who have some degree of mathematical sophistication) Grading You will be graded on the basis of one in class midterm exam, and one final exam during final exam week. Texts There will be two texts, the one that we will use up until the midterm is Mendelsohn's Introduction to Mathematical Logic. Since the first part of the book does not vary from one version of the book to any other version of the book, you may purchase any used version of any volume. For the second half of the course, you will have to purchase my notes and bring them to class regularly. The aim and Purpose of this Course The central aim of this course is to develop a firm grip on what a formal system is, and to study and prove some of the most famous and important theorems in mathematical logic, i.e., theorems about any arbitrary and rich enough system. In particular, some typical theorems that we will prove are Godel's two incompleteness Theorems, Tarski's Theory of Truth, a few paradoxes, especially concerning the Lowenheim-Skolem Theorem. We will also discuss some of the philosophical implications of Godel's Theorems, especially with respect to the Hilbert Program. Along the way we will learn notions such as recursive and primitive recursive functions and other machinery that Godel had to invent in order to prove his two famous incompleteness theorems. We will also have to study Robinson Arithmetic, and we will prove Church's Theorem as well as discuss Church's Thesis. Specific Topics and Approximate Schedule The first half of the course will consist in setting up a very elementary formal system, Propositional Logic, and prove such theorems such as that all truth functional expressions are expressible in this system, that there can be one truth functioal connective from which all other truth functional connectives can be defined, and that Propositional logic is complete. We will then introduce the semantics for our next formal system, Predicate logic. This will bring us up to the first half of the course. In addition to having to do exercises, we will be be using any version of Elliot Mendelsohn's classic textbook, Introduction to Mathematical Logic. For the second half of the course, we will learn all the important material that you will be responsible for knowing in detail on the final. We will learn Robinson Arithmetic, Arithmetization, Godel numbering, diagonalization, recursive, and primitive recursive functions, and the two Godel Incompleteness Theorems. We will be using my personal notes to cover this material. YOu will be required to bring my notes to class. Only For Graduate Students, Advanced or Interested Students wanting special Course Credit In addition to attending the regular Mathematical Logic class, graduate students and advanced or very interested honors students will meet with me one hour a week to discuss further details of Godel's two incompleteness theorems. We will also go over in more detail, Tarski's Theory of Truth and the "paradox" concerning the Lowenheim-Skolem Theorem. We will also look at the philosophical implications of the Godel Theorems on the Hilbert Program. All of these topics depend upon whether time permits.