Fall 2006 Course Offering
Math 768N (MWF 12:20 -- 1:10 PM, LeConte 316)
Models of Set Theory

Modern set theory is one of the great adventures of the human mind. The adventure began when Cantor discovered in 1873 that there is a whole heirarchy of infinite numbers, the smallest of them being the number of integers, while the number of real numbers was further up in the hierarchy. He conjectured, but could not prove, that the real numbers represented the second smallest infinite number. This conjecture, known as the continuum hypothesis (CH), had a profound influence on the future development of set theory.

This course is a natural continuation of the course I am teaching this Spring semester, Set Theory (Math 760). One of its objectives is to show how the CH is independent of the usual axioms of set theory. This means that CH can be added to the generally accepted (ZFC) axioms of set theory without leading to a contradiction, as long as the ZFC axioms themselves are consistent, and that the denial of CH can be also added to the ZFC axioms on the same basis as CH can.

This Spring, Math 760 has largely consisted of a detailed study of the ZFC axioms and showing how all of mathematics can, in principle, be reduced to the theory of sets coupled with a few simple rules of logical deduction. This follow-up course will briefly review this and then go on to the study of models of set theory, which are essential for understanding the things I've talked about in the preceding paragraph.

Models of set theory might be called ``island universes'' (the old term for galaxies). They are sets that mimic the class of all sets so closely that they satisfy every property which is generally believed to be satisfied by the universe of all sets---in other words, they satisfy all the ZFC axioms. I have been talking about them in the Seminar in Set Theory and Logic this semester, and you can read a little about them in the seminar announcements.

This Models course will not assume any prior course in set theory or logic, but it will be very demanding for anyone without prior exposure to either. Anyone who has been regularly attending the seminar this semester or who is taking 760 should have no more trouble understanding this Models course than they have understanding the seminar/760. Others may want to look closely at the seminar announcements and the link to a distillation of seminar notes in pdf form and in ps form and in dvi form.

Besides the CH, there are a multitude of other statements, not just in set theory but also in topology, algebra, and analysis, have been shown to be independent of the axioms, and this Models course will explore some of them and even show a few of them to be independent. In the links above, you can read about some statements in measure theory that are of this nature. I have also written a short expository paper with a lavishly annotated bibiligraphy, recently published electronically by Topology Atlas, that tells a little more about all this.

We will be studying all three of the usual ways of building models of set theory for independence proofs:

  • the use of large cardinal axioms, for which the seminar notes linked above are a good introduction. As in the seminar, our main goal is the understanding and uses of the concept of measurable cardinals.
  • the method of forcing. In 1963, Paul Cohen began a revolution in set theory when he introduced this method to the mathematical world. Most independence proofs are carried out using this method, which can be used to show both the consistency of CH and that of the denial of CH (which Cohen did, earning the Fields Medal for it).
  • the method of inner models, which Goedel introduced in the late 1930's, well before Cohen came on the scene, giving us the first proof of the consistency of CH. Goedel introduced a very technical concept called ``constructibility'' and showed that the class of constructible sets satisfies all the ZFC axioms and, moreover, mimics the whole class of all sets so successfully that we cannot even know for sure whether every set is constructible! And he showed that this class satisfies CH. In other words, every infinite constructible set of constructible real numbers either admits a constructible 1-1 correspondence with the (constructible!) set of natural numbers (all of which are constructible) or a constructible 1-1 correspondence with the set of all constructible real numbers.

    Of the three methods, the one of inner models is the most technical and comes last in the course. Unfortunately, there is no really good textbook for the course, so I will be using a variety of books and giving copious photocopies from them, free of charge unless someone wants a copy of a substantial fraction of a book.

    Homework will be less time consuming than it is in the Math 760 course this semester, which typically features 4 and sometimes more problems per week; this Models course will average only one problem per week, and students will get at least two chances to get the problems right: after the first collection of homework, the papers are returned with partial credit and hints on how to get it right if it wasn't right the first time. Improvements the second time around are given reduced credit: half credit if the problem involves only two chances, two-thirds credit if it involves three chances, and so forth. Any problems that are extra hard get full credit the second time also and the point reductions only begin on the next go-around.

    There will be a midterm test, but unlike the one for Math 760, it will only be counted if it improves your homework grade. This is also true of the final exam, which is optional. There will be opportunities for earning extra credit through extra difficult problems and giving lectures in seminar (like Kate Scott has done this semester) and just for attending the seminar.