# Class Announcements for Summer 2016 Independent Study

## The text

The text is here (the Math 502 notes).

Here it is.

## Here are the completeness theorem slides

The slides are here.

Read sections 5.3 (some of this is review, since it includes the definition of the object language, but it includes the definition of substitution, which is new on 7/20) and 5.4, which will be the subject of the lecture on 7/22.

I have added exercises to the counting notes. Please do at least a couple of them. If you do all of them I will be impressed.

I will add an exercise on substitutions to section 5.3 in the notes (after the exercises on parsing object language expressions which you already did): when this exercise appears in the text, do it. I may do some updates to the text of 5.3 and 5.4 while I am thinking about the lecture for Friday: I'll send you a note on email when and if I do this.

It is not a bad idea to attempt the exercises given in 5.4 as you read it.

## Notes from the 15th on sizes of sets

Here are the notes. They do not contain exercises yet, but probably will contain some on Monday afternoon.

## Assignments

Here is the first computer lab assignment.

Here is the third assignment, posted 7/1/2016 and due 7/8/2016.

## Other Announcements

The challenge problem: Prove that if {x,y) = {x,z}, then y=z. We did it in class, but its worth keeping it posted here to think about.

We are holding an additional meeting Thursday 30 July to make up for missing the 4th. Happy Fireworks! For the meeting on the 30th, read sections 3.4 and 3.5 up to page 49 and come armed with questions. Particularly notice the new definitions of union and intersection of a set: these are not the same as the usual binary union and intersection operations!

## Draft schedule

This is subject to change! Numbers are dates in July.

• 1: 3.5 more about numbers, ordered pairs
• 6: 3.6 relations and functions -- touch briefly on cardinality.
• 8: 3.9 definitions by recursion
• 11: 3.9 Peano arithmetic? Maybe a little bit about cardinality, countability, Cantor's theorem.
• 13: 3.9 simple formal languages defined by recursion (concatenation? calculator language?) Note that these are countable. Define semantics of calculator language!
• 15: 5.1 syntax of formal language of type theory (this involves some set theory tricks to be explained)
• 18: 5.1 syntax of formal language of type theory (it would be useful to discuss why the language is countably infinite if it has countably many symbols)
• 20: 5.2 reference and satisfaction
• 22: 5.2 reference and satisfaction
• 25: 5.3 formal propositional calculus (the theory of Marcel!)
• 27: 5.4 completeness theorem
• 29: 5.4 completeness theorem (compactness follows readily: Lowenheim-Skolem I cannot cover unless I have snuck in some discussion of cardinality)
• if this goes faster than expected, return to chapter 4 and talk about ordinary set theory. Also, more discussion of infinite cardinality at least to the level of Cantor's theorem would be useful.