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

- 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.

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.

- Problem Set 0, due at 6pm on Tuesday, August 30th. ps0.tex
- Problem Set 1, due at 6pm on Tuesday, September 6th. ps1.tex
- Problem Set 2, due at 6pm on Tuesday, September 13th. ps2.tex
- Problem Set 3, due at 6pm on Tuesday, September 20th. ps3.tex
- Problem Set 4, due at 6pm on Tuesday, September 27th. ps4.tex
- Problem Set 5, due at 6pm on Friday, October 14th. !CUNY-Tuesday! ps5.tex
- Problem Set 6, due at 6pm on Tuesday, October 25th. ps6.tex
- Problem Set 7, due at 6pm on Tuesday, November 1st. ps7.tex
- Problem Set 8, due at 6pm on Tuesday, November 8th ps8.tex
- Problem Set 9, due at 6pm on Tuesday, November 15th ps9.tex
- Problem Set 10, due at 6pm on Tuesday, November 22nd ps10.tex

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.

- 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.

- 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)

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.

If \(f\) is injective, then it is surjective.

- 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);

- 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.