Class Announcements Page for Lauren Stevens Math 496 Independent Study, Spring 2015

Here is your class announcements page. Keep an eye on it. We have agreed on meeting times 11 AM -- 12:30 pm Mondays and Thursdays, at my office, Mathematics building room 240A.

The Independent Study Application

Here is the independent study application which defines the course.

The Text

Here is your text. The link at the top of my home page does not lead to the same file: edits to this file in the course of our independent study will not necessarily be posted immediately to the public link.

Last Assignment (due in final exam week)

Prove that the Kuratowski pair {{x},{x,y}} has the basic property of pairs (x,y)=(z,w) -> x=z and y = w.

Prove the theorem of Zermelo set theory that for any A, {x E A : not x E x} is not an element of A.

Since I know that you are not going to look at these until later, I feel free to add another exercise or two during this coming weekend -- I will look over the hard recent material and see if there is something which I think is worth you attempting.

3/19/2015 Instructions

I'm going to assign you some Things to Do (or attempt): first, try to prove the associative law of multiplication in Peano arithmetic.

Second, do problems 1,4,5 in the homework set on pp. 93-4. Problem 4 is to recapitulate a proof I did in class; good notes will make it easy. In problem 1, you might have to draw pictures to indicate how some bijections are computed (or you might not know how to do them of course -- don't hesitate to ask me). Problem 5 should be quite similar to (and easier than) examples I did in our meeting.

This is all due Thursday after break.

3/17/2015 Instructions

Ill keep these simple. Prove that the relation ==P determined by a partition P of A defined by x ==P y iff there is B in P such that x is in B and y is in B is an equivalence relation on A (you can see this stated in proper notation on p. 83 in the middle of the page (as a Theorem, with the proof left as an exercise). I'll have more homework for you on Thursday for you to work on during the break. My main hint about this one is, unpack the definitions and make sure you prove everything you need to prove!

February 26th Instructions

Please write out the proof that the set of natural numbers is inductive in full detail and hand it in at our next meeting.

Please read the selections from the Hoftstadter book and be ready to comment at the next meeting: I will be talking about similar things. Also, be ready to talk to me about universals; I would like to know what you as a philosophy student have learned to say about these.

There is also a small assignment, due on Thursday March 5: please do the exercise in section 3.7.1, currently on p. 70. This exercise is about unpacking definitions and using logic, stuff that you can do. Hints will be offered if required.

February 9th Remarks

Read the section on finite numbers (and unions and intersections of sets) to understand the development on pp. 51--54 culminating in the proof that the Axiom of Infinity is equivalent to Peano's Axiom 4. I will lecture this on Thursday in a different and hopefully instructive order (giving the motivation, which the current presentation does not unless you read it backward, which is quite typical of mathematical text). My intention is to begin with the statement of Axiom 4 and try to prove it, discovering by seeing where the argument breaks the need for the Axiom of Infinity and for the specific lemmas that I prove.

A note: I managed to precisely reproduce my argument for homework problem 3 using Marcel, and it would be useful for you to see what the additional feature is (the cut rule).

Assignment III

Here is the third paper homework assignment. The due date is self-adjusting in case of difficulties. In fact, I think the due date is optimistic and I do expect to be giving guidance on these problems on or before Friday.

Lab II and Assignment II

Here is the second computer lab, due 2/2/15. Here is the second paper homework assignment, also due 2/2/15. Both due dates may be flexible if you run into difficulties.

Here is the text file with the start command for the first problem in lab 2.

Jan 16 Instructions

In Principia Mathematica, read the introduction to the first edition until you get into trouble (p. 1 etc). Read section A of the main text (starting p. 90) until you get into trouble (at the very least identify the axioms and rules of propositional logic and write them in our notation), with any additional comments that may come to mind. How much we do with this will depend on your interest and effort. Read Quine, "On What There Is". What do you think?

On Friday we will formally talk about quantifier rules and start a quantifier computer lab. For Friday's meeting, read sections 2.7-2.10 from our text and be ready with your questions and confusions...

Lab I and Homework I

Here is your first computer lab, due by class time Friday January 23rd. This is edited from a lab I gave to a Math 314 class in 2011; I hope I have removed all the anachronisms!

Here is your first paper homework assignment, also due by class time Friday January 23rd.

Be aware that if you are taking too long to do an assignment, you can ask me for additional time.

Our First Meeting Jan 12 2015

I'm going to be bold about the reading assignment. Read sections 1.2 to 2.6; be aware that we are not really reading these sections and going past them in one day! Bring your confusions if any to the first meeting... I'm thinking that since you have already taken a course in formal logic, the material will not be unfamiliar, but my approach may be so.

I am also interested in hearing from you on the first day about your aims in taking this course; be ready to talk about what you think you know in this area and what you want to find out about. As we start the first unit on proof techniques, I will be interested in hearing from you about how you perceive my formal treatment as differing from what you learned in your courses in the Philosophy department.

Please bring your laptop computer if you have one.