Forcing and Independence in Set Theory

Instructor: Sherwood Hachtman

Lectures: 11am-1pm in MS 6201

Problem-solving sessions will be held in

Lecture Notes by Spencer Unger.

Exercises:

Day 1: Well-orders, cardinals, cofinality. Here are Hints

Day 2: Cardinal characteristics, etc. A hint

Day 3: Posets, dense sets, antichains and MA.

Day 4: MA and cardinal characteristics, Lebesgue measure.

Day 5: MA and ultrafilters; the club filter.

Weekend 1: An assortment of challenging problems for the weekend. Some hints

Day 6: MA and trees. A hint

Day 7: First Order Logic.

Day 8: Models of Set Theory. Hints

Day 9: More models, ultraproducts, forcing: names and filters.

Day 10: Forcing: some practice with names.

Weekend 2: Some extra weekend problems.

Day 11: Forcing facts; Automorphisms; Cohen forcing.

Day 12: Almost homogeneous posets, forcing equivalence, distributivity.

Day 13: Projections, forcing and trees.

There are certainly moments in any mathematical discovery when the resolution of a problem takes place at such a subconscious level that, in retrospect, it seems impossible to dissect it and explain its origin. Rather, the entire idea presents itself at once, often perhaps in a vague form, but gradually becomes more precise. Since the entire new 'model'M(a) is constructed by transfinite induction on ordinals, the definition of what is meant by sayingais generic must also be given by a transfinite induction. Yeta, as a set of integers, occurs very early in the rank hierarchy of sets, so there can be no question of buildingaby means of a transfinite induction. The answer is this: the setawill not be determined completely, yet properties ofawill be completely determined on the basis of very incomplete information abouta. I would like to pause and ask the reader to contemplate the seeming contradiction in the above. This idea as it presented itself to me, appeared so different from any normal way of thinking, that I felt it could have enormous consequences. On the other hand, it seemed to skirt the possibility of contradiction in a very perilous manner. Of course, a new generation has arisen who imbibe this idea with their first serious exposure to set theory, and for them, presumably, it does not have the mystical quality that it had for me when I first thought of it. How could one decide whether a statement aboutais true, before we havea? In a somewhat exaggerated sense, it seemed that I would have to examine the very meaning of truth and think about it in a new way.— Paul Cohen inThe Discovery of Forcing, Rocky Mountain Journal of Mathematics, Vol 32, No 4, 2002.