# Elements of Set Theory 2: An introduction to large cardinals, spring 2013

MAKEUP CLASSES ARE SCHEDULED AS FOLLOWS: Tuesday May 7TH, 12-14, Tuesday May 14TH, 10-12. Rooms both days: CK107

Lecturer

### Scope

10 cu.

### Type

Advanced studies.

### Prerequisites.

This is the continuation of the course Elements of Set Theory given last fall. Note that our final theorem of the last semester was 7U of Enderton's book. In the spring semester we will pick up right at this point or even well before, finish Enderton and then go on to Drake's book---large cardinals, ultraproducts etc. I.e., we begin by refreshing our memory about the basic theory of cardinals and ordinals, then we move on to large cardinals, e.g. inaccessible, weakly compact, measurable etc; trees, Levy Hierarchy and the Reflection Principle, and finally constructibility. Prerequisites for the course are a reasonable background in mathematical logic, for example as covered in the mathematical logic course given in the department last fall.

### Lectures

Weeks 3-9 and 11-18, Monday 10-12 in room C129 and Tuesday 12-14 in room B321. Two hours of exercise classes per week.

Easter holiday 28.3.-3.4.

As of January 29th we defined the Levy Hierarchy, from p. 76, chapter 3 in Drake. We also finished showing that V_kappa for kappa inaccessible, is a model of ZFC. As of February 12 we covered the sections in Drake on Absoluteness and started section 5, in chapter 3 on coding satisfaction. Friday we will talk about Levy-Montague Reflection. As of Februar19 we got up to theorem 1.10 on page 130 of Drake. We will sh0ow next why the CH and the AC hold in L. As of February 26th we proved that the CH holds in L. This involved Skolem functions, the operator HC(x), and the strengthened version of Levy Reflection, theorem 7.4 on page 104 of Drake. Next we will move on either to forcing, or to other topics in Drake, depending on students' preference. As of March 12 we proved that the principle Diamond_0 implies the CH. Then we covered the material in Kunen on partial orders, filters, dense sets, generic filters. As of March 19 we covered chapter 5 of Kunen up to the definition of forcing on p. 194, i.e. definition 3.1.

### Exams

There will be one final exam at the end of the course. You can substitute problems proved in class for 4 out of 8 exam questions. Speak to the instructor for details.

### Bibliography

Course is based on the text by Frank R. Drake, "Set Theory: An introduction to Large Cardinals". WE ARE NOW USING KUNEN'S BOOK "SET THEORY: AN INTRODUCTION TO INDEPENDENCE PROOFS" A copy of the text is now on the 3rd floor in a notebook next to the other course material.

### Registration

Did you forget to register? What to do.

Homework Tuesday January 15th:

Prove Theorems 7U, 7V, 7W, 7X, starting on p.203. Prove Hartog's theorem on p. 195. Do exercises 15, 18 on page 194. KEEP REVIEWING THE MATERIAL IF YOUWERE NOT IN CLASS LAST FALL. You can do this by going to the webpage of Elements of Set Theory 1 and doing all the exercises. For January 29th also, p. 207: 26, 27.

Homework Tuesday February 12th: Prove Koenig's theorem. Does this theorem place any constraint on the value of the cardinality of the continuum? Why or why not? Prove: for any ordinal lambda, cf(lambda) is a regular cardinal. Prove: If alpha and beta are ordinals and alpha can be cofinally mapped into beta by a 1-1 function, then alpha and beta have the same cofinality. (Added later: NOT TRUE! Find a counterexample for homework Feb. 26.) Prove: successor cardinals are regular. p.257 Enderton, problems 9,11. Homework for Tuesday February 19: problem 6, p. 89 of Drake (all homework problems are from Drake from now on). problems 1,2,3,4 on p. 81. Homoework for Feb. 26: p. 81, 1,2,3,4. p. 97, 5. Homework for March 13: Prove that "y=Aleph(x)" and "z= {f| f: y --> x}" are not Sigma 1. Hint: use corollary 7.5 on page 104. Prove theorem 7.3 on page 65. Do exercise 13, p. 43. Homework for March 20th: Ca you arbitrarily add any real number to a model of set theory? Why not? p 86 of Kunen, exercises 1, 4. TC(x) is pi_1 (in fact it is Delta_1). Why? y=tc(x) iff y is the smallest transitive set containing x, i.e.

[(for all z(if z is transitive and x is in z, then y is a subset of z)) AND (y is transitive and x is an element of y)]. Homework for March 27th: p. 89 Kunen, problem 28 (Cantor's isomorphism theorem). Why is MA(omega) true? Why is MA(2^omega) false? (You can read about Martin's Axiom starting on p. 54 of Kunen.) TRY: p. 86 (Kunen), problem 1. The Delta-lemma is very important! For homework on April 9th Ivan will prove that the axioms of choice and replacement hold in M[G]. This is thm. 4.2 on p. 201 of Kunen. Your homework assignment is to read these proofs in advance...Homework 23.4: Prove that cardinals don't "collapse" if the forcing notion has the ccc. Prove that the Replacement axiom holds in M[G]. (This is proved in the text but prove it yourself. ) Prove that the partial order F consisting of all finite partial functions from omega --> {0,1} has the ccc. (So F={f|f:A–>{0,1}, A is a finite set of natural numbers}.) Prove that the CH holds in L, in outline. Prove that if kappa is weakly inaccessible in M, and G is P-generic, where P is a partial order with the ccc, then kappa remains weakly inaccessible in M[G]. Homework for April 30: Prove lemma 4.5 (a) on p. 204 in detail. p. 238: A5, A6, A9. p.246: G5.

### Exercise groups

Group | Day | Time | Place | Instructor |
---|---|---|---|---|

1. | Tuesday | 14-16 | CK107 | Ivan Chuppin |