Time: Monday/Wednesday, 3:00-4:15 pm
Instructor: Kenny Easwaran
Office Hours: By appointment
There are three main skills that I want to develop through this class. The two most important ones are the skills of being able to write up a mathematical argument, and being able to present a mathematical argument verbally. To that end, students will be asked to write up one mathematical proof each week, and to present proofs in front of class. These will be low-pressure presentations, with audience members being encouraged to ask questions, and to help out, as students work through the material together. There will be no other graded assignments, though I will give a few quizzes to help figure out what material we should spend more time reviewing. Because of the importance of in-class presentations, students should let me know in advance if they expect to miss class, and should meet with me or another student to make sure they understand the work that happened in their absence.
The third main skill I want to develop in this class is an understanding of the core ideas of mathematical logic, and particularly the theorems of Cantor, Tarski, and Gödel. These results form the foundation of the theory of computation, and are important to much 20th century work in the philosophy of language, mind, and mathematics, among other areas. Importantly, these results help us to understand how abstract formal reasoning works at all when it does, and also help us understand what sorts of things that abstract formal reasoning cannot do. Most students won't be able to recall the details of the proofs of these results years from now, but having worked through the details at least once will give you a good basis for revisiting them (or other mathematical results) when they might become relevant in your future careers.
The textbook is Boolos, Burgess, and Jeffrey, Computability and Logic but it will be supplemented by my own notes. There are many other texts that supply relevant material and I can point you to the relevant chapters of each.
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Sizes of Infinity
Boolos, Burgess, and Jeffrey: Chapters 1 and 2
The soundness and completeness of deductive logic
Boolos, Burgess, and Jeffrey: Chapters 9 and 10
The essential incompleteness of arithmetic
Boolos, Burgess, and Jeffrey: Chapters 15, 16, and 17