# Fall 2013 Math A4400: Mathematical Logic

Course Meeting: MW 2:00-3:40 PM, in room 6113 of North Academic Center Section: EF Code: 4555
Instructor: Alice Medvedev
Office: 6278 NAC
Office Hours: Monday 4-5(this course), 11-12(calculus), or by appointment.
E-mail: amedvedev at ccny

#### Problem sets:

You may work together on homework problems (and I encourage you to do so), but you should write your solutions yourself and acknowledge that you have worked together, i.e. write on the solutions you hand in "I worked with John Lee on problem 3, and with Jose Rodriguez on problems 2 and 4." Similarly, if you use any sources other than the textbook for the course, give a traceable reference to your source(s); "wikipedia" or "a theorem in a number theory book" is not traceable; "the wikipedia page for Equivalence_relation" and "Theorem 3.7 on p.54 of Burton's Elementary Number Theory" probably are traceable.

I also encourage you to use LaTeX to typeset your homework solutions. Latex is used to typeset anything with formulas by almost all scientists and some engineers, so it is worth learning whether or not you intend to become a mathematician. It is fairly easy to learn by example, and less easy to learn from books and tutorials, so I will post the source-code (.tex files) for my homeworks for you to use. If you do not want to install anything on your computer, there are many online-compiling options.

There will be weekly problem sets, due at the beginning of class Wednesdays. There will be two midterm exams, on Monday, September 23rd and on Wednesday, October 30th. The final exam for this course is held on Wednesday, December 18th, 1:00pm–3:15pm in our usual room (I think?).

The sum of all the homework grades and the sum of the three exam grades will have equal weight in determining your course grade. The raw homework and exam grades are not percentages to be converted into letter-grades as in high-school! If you are not sure how you are doing in the course, talk to me.

### Basics

Description: A first course in mathematical logic. Here you will meet syntax and semantics of (mostly) first-order logic, with proof theory conspicuously neglected and examples from other branches of mathematics emphasized. The goal of this course is to see familiar mathematics from a new perspective, thereby acquiring new and powerful tools like the Compactness Theorem of first-order logic. If time permits, we might get a chance to look at recursion theory and Godel's Incompleteness Theorems; it is quite unlikely that we'll have time to do this thoroughly.

Prerequisites: An interest in formal logic and the ability to write proofs.
Abstract algebra is an official prerequisite for two reasons. There will be many algebraic examples in this course. More importantly, you are expected to be able to write clear, precise proofs.

Texts: The main text for the course is Fundamentals of Mathematical Logic by Peter G. Hinman. This text may be a bit too terse; I will supplement it with many examples and explanations in lecture. You might find A Mathematical Introduction to Logic, Second Edition by Herbert B. Enderton easier to read, but beware subtle differences in definitions and notation!
I do not provide lecture notes or homework solutions - but I am happy to answer your questions during office hours. Should some of you wish to supply such notes or solutions for your fellow students, I will be happy to make space for them on the blackboard website, or even scan them and post them myself.
I am not sure yet to what extent I intend to use blackboard.

Content: We'll cover propositional logic (sections 1.1-1.4 of Hinman) fairly quickly. We'll cover the basics of first-order logic (sections 2.1-3.1 and 3.5) thoroughly. We'll discuss chapters 4 and 5 (Godel's Incompleteness Theorems) in some level of detail, depending on time.
I will follow the book's notation quite closely. I will often present material in a different order from the book, in an effort to present semantics ("meaning") before syntax ("grammar").

### Beginning of the course:

If you have not had much experience with abstract mathematics, Introduction to math notation by George M. Bergman will help. You also want to make friends with equivalence relations (for example, here or here) and with basic naive set theory (Chapter 0 of Enderton).

### What is a proof?

Most of the problems in this course will ask you to "prove" something, that is, to give a convincing explanation of why this something is true. At best, your proofs should look like the ones in your textbook. In particular:
• Proofs are made of complete, grammatically correct sentences.
• All variables that appear in the proof either appear in the statement being proved, or are clearly introduced somewhere in the proof with a "let".
• Each statement either clearly, logically follows from previous statements (and that logic is explained), or is introduced with an explicit purpose (e.g. "suppose towards contradiction that..." or "the inductive hypothesis is...")
• Anything that is not proved is cited, by its common name (e.g. the Fundamental Theorem of Arithmetic) or by reference to our textbook (e.g. Proposition 4 on p. 10)
The point of the proof is not to demonstrate to the grader that you understand the ideas in the problem, but to explain the solution to someone else in the class who has not thought about this particular question, and to whom you can only write, not speak. In particular, what I write on the board during lecture does not constitute written proofs - it is quite meaningless without the things I say!

### Etiquette and Attitude

These are things that most professors take for granted; some are more obvious than others; some are more important than others; many are equally standard outside academia.

Communication. Written communication, such as emails, begins with a greeting, probably addressing the recipient, and ends with a signature, probably including the full name of the writer. You have no idea how much some professors are offended by informalities such as "Hey," and "See ya."

It is reasonable to expect emails to be answered within a day or two; it is unreasonable to expect an answer within an hour. The truly urgent questions (where's the final exam - i'm already late?) are better answered by google or a phone call to the relevant university office. Some questions that feel urgent (did I pass the class?) simply require patience.

Email is great for logistics: finding a time to talk outside regular office hours, making special arrangements for missed work, telling me about a problem with the web page or a homework, etc. Email does not work well for discussing mathematics - come to my office hours instead.

If you a professor (or, really anyone else!) agrees to meet with you personally, outside of lecture and standard office hours, and then you find out that you will not make it to the meeting, you should inform the professor of this at least several hours in advance. I have had enough problems with this issue that I will take 1 percentage point off your grade for each missed appointment.