Fall 2013 Math A4400: Mathematical Logic
Course Meeting: MW 2:00-3:40 PM, in room 6113 of North Academic Center
Instructor: Alice Medvedev
Office: 6278 NAC
Office Hours: Monday 4-5(this course), 11-12(calculus), or by appointment.
E-mail: amedvedev at ccny
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.
- Problem Set 0, due at 2pm on Monday, September 9th.
- Problem Set 1, due at 2pm on Wednesday, September 18th.
- Problem Set 2, due at 2pm on Wednesday, September 25th.
- Problem Set 3, due at 2pm on Wednesday, October 2nd.
- Problem Set 4, due at 2pm on Wednesday, October 9th.
- Problem Set 5, due at 2pm on Wednesday, October 16th.
- Problem Set 6, due at 2pm on Wednesday, October 23rd.
- No homework due Wednesday, October 30th, the day of the second midterm.
- Problem Set 7, due at 2pm on Wednesday, November 20th.
- Problem Set 8, due at 2pm on Wednesday, December 11th.
Assignments and Grading:
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.
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.
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.
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) 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:
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!
- 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)
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.
Responsibility. On the most basic level, it is your responsibility to know about all assignments and deadlines and to show up for exams, and to make any special arrangements necessary, from arranging time to meet outside office hours to knowing when your drop deadline is.
You are also responsible for your learning, from knowing what material has been covered to making sure you understand that material. The lectures will not cover everything; the problem sets will not cover everything; the textbook is the closest to covering everything. You will certainly have to learn some things on your own, either from the textbook or really on your own. Some of the material this course covers is genuinely difficult: you will probably not get it on the first try; that's ok. To maximize the number of tries, read the textbook before the lecture, noting the parts you don't understand; then come to lecture and pay special attention to those parts and ask lots of questions; then read through the book again and you will discover new subtleties you don't quite get. Come to office hours to clear up those, work on the homework, come to more office hours, read comments on graded homework, and read through the text one more time. By this point, two or three weeks after meeting the concept for the first time, you should be comfortable with it. The bottom line is, your job is to acquire knowledge, not simply to follow my instructions. This course is graded on accomplishment, not effort.
Scholarship. Whatever you do with the rest of your life, in this course you are acting as a mathematics scholar, grappling with ideas that are new and confusing to you. Thus, your natural state is confusion: the moment you understand something, you move on to the next topic. Progress is measured by being confused about different ideas over time.
So, what do you do with a confusing definition or theorem or proof? First, you make sure you understand each word and symbol that appears in it. The "appear in" partial order is well-founded, so induction works. Then you try to think of explicit examples. For a definition, look for things that satisfy it, as well as for things that don't. For a theorem, look for an example that satisfies the hypothesis and see that it satisfies the conclusion, and then for some examples that don't satisfy the conclusion and see which hypotheses they fail. Similarly, to understand a proof, it is often helpful to go through it with a specific example in hand; a non-example that fails some of the hypotheses will make it easier to see where the proof needs those hypotheses, i.e. breaks without them. The best way to understand a proof is to completely forget it and then try to prove the result yourself; this is also the slowest way. The next step is to ask why this particular definition was chosen, or why the theorem was stated in this particular way. Varying a definition, you may find equivalent definitions, stupid definitions, or new interesting concepts. Varying a theorem, you will find many false statements, some true generalizations, and occasionally an entirely new result.
This is what us mathematicians do when we aren't teaching.
Writing. This is a writing-intensive course. Almost all problems on homeworks and exams require explanation and justification; these should be written out in words, in complete sentences. Ideally, your writing should closely resemble the proofs and examples in our textbook.
Cheating. Please don't. If you are writing down ideas somebody else told you and not mentioning this fact, you are cheating. If you are writing down words somebody told you, and you could not rewrite the solution in different words or explain it to another student in the class, you are cheating. Rules around exams are not arbitrary hurdles I place in your way. They are something like arms-control deals amongst the students in the course: you really don't want infinite-time open-everything take-home exams, because those require, well, infinite time and access to everything.
Letters. If you need a letter of reference of some sort, make sure to ask for it long long in advance: at least a month before any deadlines, and in any case no later than mid-November.
Lectures Attendance is not mandatory but strongly recommended. Most of the material covered in lecture is in the textbook (though sometimes in much less detail), and most of the announcements will eventually be posted on this course webpage. However, you are responsible for the exceptions, so if you miss a class, talk to someone who didn't. You will also find the class much much harder if you do not come to lecture; in my experience on both sides of the game, missing more than half of the lectures will cost you about two letter grades.
The best use of lecture is to focus all of your attention on it, for the entire duration. If you are not doing this, please do not disturb the people who do. If you come late or leave early, sit near the door and don't let the door slam. If you're eating, please don't be loud or smelly (if you bring hot pizza, bring enough for everyone!). Whether or not you're playing with your cell phone, make sure it's silent. Basically, if it's distracting, don't do it. Common sense, right?
The best way to stay focused in class is to get involved. If something doesn't quite make sense, ask about it! I like questions! I like stupid questions, too - for every brave soul willing to ask one, there's ten shy confused students thinking the same thing.
If you think there's a typo on the board, you may well be right - ask about it! Sometimes, I'll make them on purpose, to keep you on your toes. Other times, I'll make honest mistakes - I'm not perfect. And if it's not a typo, your confusion will not get cleared up if you don't ask. Clearing up confusions is what this is all about.