# Fall 2016 Math 711: Logic

When: TuTh 6:00PM-7:30PM Where: CUNY Graduate Center room 5417
Course Code: 32470
Course webpage: "http://www.sci.ccny.cuny.edu/~abear/math71100f16/coursepage.html
Instructor: Alice Medvedev
Office Hours: Th 5-6pm in GC room 5417
E-mail: medvedev math ccny at google

## End-of-semester stuff.

• The final exam will take place on Tuesday, November 13th, 6-7:30pm, in room 5417 at the CUNY Graduate Center.
• You may resubmit new solutions to previously graded problem sets for up to 3/4 credit.
• Here are few more problems to work on in preparation for the final exam. review.tex
• If you write up and submit your solutions to these review problems, I will will grade them and replace your median problem set score, if and only if that helps your grade.
• The absolute final deadline for submitting work for this course (late homeworks for half credit, rewrites, review sheet solutions) is 6pm on Tuesday, November 13th.

## 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 textbooks 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 typeset your homework solutions with LaTeX: you will use it to typeset your PhD thesis, and almost all scientists and some engineers use it as well. 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.

## Course organization and content.

Description. This is a first course in mathematical logic. Here you will meet syntax ("grammar") and semantics ("meaning") of propositional and 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. Hopefully, we will also have time to examine the first-order theory of algebraically closed fields in some detail.

Prerequisites. General mathematical maturity: problem-solving, proof-writing, and comfort with abstraction.
Chapter 1 of van den Dries is a good summary of mathematical concepts and notation that I will use without explanation; here is another concise explanation of mathematical notation and language; and here is a concise introduction to equivalence relations with many examples.
While there will be many algebraic examples in this course, abstract algebra background is not strictly necessary. Dummit&Foote is the best abstract algebra textbook and reference.

Books. We begin with Mathematical Logic Lecture Notes by van den Dries (click link or download from the author's webpage) for propositional logic. By mid-late September, we shall switch to Rothmaler's Introduction to Model Theory (Gordon & Breach 2000). Many examples and exercises will be drawn from Hodges' Shorter Model Theory (Cambridge University Press 1997), a wonderful Model Theory book that you don't need to own just yet.

Content. We shall thoroughly cover Chapters 1-3 of Rothmaler and Sections 2.1-2.6 of van den Dries. We shall prove the Compactness (aka Completeness, aka Finiteness) Theorem for first-order logic in one or two ways, and learn to use it by working through many of its applications. If time permits, we may cover additional topics, such as model-theoretic proofs of Hilbert's Nullstellensatz and Chevalley's theorem, also known as quantifier elimination for the theory of algebraically closed fields.

Calendar.
• Part I, 11 classes: Thursday, August 25th to Thursday, September 29th.
• Two-week New 5,777th Year break.
• Midterm Exam on Thursday, October 13th.
• Class on fake-Tuesday October 14th!
• Part II: 11 classes: Tuesday, Ocotber 8th to Tuesday, November 22nd.
• Part III: 4 classes after Thanksgiving.

Part I, the foundation of the course, will cover sections 2.1 - 2.5 of van den Dries: propositional logic, and basic syntax and semantics of first-order logic. Chapters 1 and 2 and Section 3.1 of Rothmaler cover the same material as van den Dries 2.3-2.5. The midterm exam will cover this material.

Part II, the heart of the course, will cover Chapters 3 and 4, and some of chapter 5, of Rothmaler; with a possible detour to a completely different approach to the same topic in Sections 2.7, 3.1, and 3.2 of van den Dries. We'll deepen our understanding of first-order logic to state, prove, and use the Compactness Theorem, also known as the Finiteness Theorem (see Rothmaler), and closely related to the Completeness Theorem (see van den Dries).

Part III, the icing on the cake, will consist of some advanced topics around the theory of algebraically closed fields, such as the model-theoretic approach to Hilbert's Nullstellensatz and Chevalley's theorem; the full proof of Ax's Theorem; and/or strongly minimal sets in general.

• 40% weekly problem sets due on Tuesdays at 6pm;
• 30% midterm exam on Thursday, October 13th;
• 30% final exam.
If there is sufficient interest, the final exam may get replaced by a collaborative final project.
Every time your technology (cell phone, laptop, et cetera) makes a sound during class, you will lose 1% of your semester grade.

## My background assumptions.

### 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 a professor (or, really, anyone!) 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.

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.

## Propaganda.

Theorem (Ax): Let $$f: \mathbb{C}^3 \rightarrow \mathbb{C}^3$$ be the function $$f(x,y,z) := (p(x,y,z), q(x,y,z), r(x,y,z))$$ for polynomials $$p, q, r$$ with complex coefficients.
If $$f$$ is injective, then it is surjective.
Proof: Logic!

Logic disects and analyzes the language of mathematics, and thereby
• resolves ambiguities of natural language;
• identifies similarities and formalizes analogies across mathematics, somewhat like category theory, but not exactly.
• proves new theorems in algebra, geometry, number theory, et cetera (Ax's Theorem);
• proves that some questions are unanswerable (Hilbert's 10th Problem);
This course will
• give you a new perspective on math, especially all things algebraic;
• start you on the path towards research in mathematical logic (model theory, recursion theory, set theory, proof theory, reverse math);
• help you to write clear proofs and to teach proof-writing and reasoning;
• (with Math 712 in Spring) prepare you for the Logic Qualifying Exam.