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Math 160 Syllabus (Mathematical Logic) Winter 2003, UCSC Here's why you should take this course: Statement 1: Both of these two statements are false. Assuming that a statement is either true or false (and not neither or both), what happens if the first statement is true? What does this imply for the second statement? Little puzzles like the above have profound consequences when one tries to get to the bottom of notions like "truth", "provability" and such. One of the great strengths of Mathematics is that once you have a proof of a statement, it becomes certain, or irrefutable. But what does this certainty rest on? Any proof needs assumptions. Are there universal truths that can serve as those assumptions? Take an example from euclidian geometry: Many proofs depend on the assumption that parallel lines do not intersect. But can you, and is so, how do you prove that? Can we be sure that we will never be able to prove incorrect statments like "0=1"? Can we possibly even be able to prove that? Does every mathematical question have an answer in the form of a proof? As an example, can we be sure that a proof or counterexample to Goldbach's conjecture can be found? Could it be that one day a clever programmer invents a machine (theorem prover) that can answer all mathematical questions? We will adress these issues and hopefully have a lot of fun tracing some of the most interesting and intriguing developments in mathematical logic during the last century. Your Teachers
Class Meetings
Course Goals Our goal is Goedel's First Incompleteness theorem. To get there, we will cover selected topics of Chapters 1,2 and 3 from the textbook with as much detail as time allows. Chapter 1 deals with propositional calculus (sentential logic). This should be familiar to you. We will follow the tradition of introducing the main concepts in this arena which is more readily understood (and where it is a lot easier to prove things.) Chapter 2 discusses the main principles of predicate calculus (first-order logic). We will cover the basic definitions, consequences and the soundness, completeness and compactness theorems in detail. Chapter 3 is about the limitations of the system. Here we will in detail trace Goedel's ideas that lead to the proof of his incompleteness theorem: If number theory makes sense, then for any finitely axiomatized system for number theory there are true facts that cannot be proven in this system. We will cover as much detail of the proof as our time allows. We will also discuss the meaning and historical importance of this theorem as well as some further incompleteness results. Requirements To be eligible to enroll in Math 160, you must have taken and passed Math 100 or CSE 101. Talk to me if you have any questions about that. Title: Introduction to Mathematical Logic Available for purchase at the Bay Tree Bookstore. Exams & Grading Policy There will be a midterm (30%) and a comprehensive final (40%). Weekly homework assignments are also a big part of your grade (30%). Other factors such as participation in class and section may also affect your final grade. Homework Homework is assigned at the end of each lecture from the sections covered. Chris is in charge of all matters homework. In particular, hwk will be collected and returned in section. See the class web site at http://count.ucsc.edu/~bauerle/Default.htm for the list of problems and other information. Collaboration Collaboration is encouraged in working on problems, but is not allowed on exams. Therefore it is important (and required) that each student turn in their own homework assignments. |
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Bäuerle, Ph.D., UC Santa Cruz.
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