Appalachian set theory

Todd Eisworth and Justin Moore

"Iterated Forcing and the Continuum Hypothesis"

A special two-day Appalachian set theory workshop will be held at the Fields Institute in Toronto, Canada, on May 29-30, 2009

Approximate schedule on Friday and Saturday

8:45 - 9:30 morning refreshments at the Institute
9:30 - 12:30 lectures with breaks at convenient times
12:30 - 14:30 break for lunch (not organized)
14:30 - 18:00 lectures with breaks at convenient times


Do you have the documents needed for travel to Canada?


There is no registration fee. By registering in advance, you help us plan the workshop; for this, go to the Fields registration webpage. On-site registration will also be available.

Mathematical overview

In 1965, Solovay and Tennenbaum [JSTOR] introduced the technique of iterated forcing in their proof of the consistency of Souslin's Hypothesis (SH). Since then, iterated forcing has assumed a central role in establishing consistency results. The associated technology has grown increasingly sophisticated and, many times, advances in set theory have been fueled by corresponding breakthroughs in iteration theory. Although we know much about iterated forcing, there are still many general questions, and in this workshop we address one such:

Why is it so difficult to use iterated forcing to produce "interesting" models of ZFC + CH?

What sorts of difficulties arise? Consider, for example, the problem of obtaining a model of CH in which SH holds. Recall that a Souslin tree is an ω1-tree with no uncountable chains or antichains, and SH says there are no such trees. Early on, Jensen showed that the combinatorial principle "diamond" implies there is a Souslin tree. Given a Souslin tree, forcing with the tree turned upside down adds a generic chain through the tree, thereby making it non-Souslin. This forcing does not add reals, so it does not collapse ω1 and the tree remains an ω1-tree. A natural way to proceed is to start with a model of GCH and iterate killing Souslin trees, those in the ground model, as well as those that arise in intermediate models, until none are left. This procedure yields a model of SH. One might hope, because we are not adding reals at successor stages, that CH is preserved by the iteration. However, Jensen showed this naive approach runs into problems in that new reals may appear at limit stages of the iteration. A large part of the monograph by Devlin and Johnsbraten [MathSciNet] is devoted to an exposition of how Jensen overcame these issues to produce a model of ZFC + CH + SH.

Things have progressed significantly since Jensen's work in the late 1960s. For example, a close study of the way in which iterated forcing with Souslin trees can add reals culminated in the isolation of "weak diamond" by Devlin and Shelah [SpringerLink]. In 1984, Shelah's seminal monograph, Proper Forcing [MathSciNet], introduced the notion of D-completeness, a key ingredient in proofs that certain iterations do not add reals. Many other tools for preserving CH in iterations have been developed, but the new methods are complex and understood by only a handful of experts. This workshop is intended to remedy the situation.

Our workshop will be organized around two guiding principles. First, we want to give attention to examples of "single-step" forcings, that is, partial orders that accomplish a particular task without adding reals. Second, we wish to give an account of how iterations can be done without adding reals. Many examples will be presented, and many open questions will be formulated --- the talks will be pedagogical, and not merely the reporting of results. Our intent is that participants should leave with a firm understanding of how the basic iteration theorems work, in addition to acquiring a reasonable knowledge of the current state of affairs.

Further reading

Basic graduate level textbook on set theory:

General information on iterated forcing and not adding reals:

Examples of how iteration theorems for not adding reals are applied:

An example of a proof of an iteration theorem for not adding reals:

An example of how reals can be added in an iteration:

Participant travel support

Funds provided by the National Science Foundation and the Fields Institute will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to Ernest Schimmerling by email.

Lodging options

Related activities

A few workshop participants who have arranged slightly longer visits to Toronto will speak on other days:

Post workshop materials