Appalachian set theory

# "Iterated Forcing and the Continuum Hypothesis"

## 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

## Attention

Do you have the documents needed for travel to Canada?

## Registration

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:

• Kenneth Kunen, Set theory; An introduction to independence proofs, North-Holland

General information on iterated forcing and not adding reals:

• Uri Abraham, Proper forcing, to appear as a chapter in the Handbook of Set Theory [PDF]
• Saharon Shelah, Proper and improper forcing, second edition, Springer-Verlag

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

• Uri Abraham and Stevo Todorcevic, Partition properties of ω1 compatible with CH, Fund. Math. 152 (1997), no. 2, 165-181 [PDF]
• Justin Tatch Moore, ω1 and -ω1 may be the only minimal uncountable linear orders, Michigan Math. J. 55 (2007), no. 2, 437-457 [Project Euclid]

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

• Todd Eisworth and Peter Nyikos, First countable, countably compact spaces and the continuum hypothesis, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4269-4299 [AMS]

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

• Keith J. Devlin and Saharon Shelah, A weak version of diamond which follows from 2aleph0 < 2aleph1, Israel J. Math. 29 (1978), no. 2-3, 239-247 [SpringerLink]

## 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.

• Your name, university affiliation, mailing address, phone number and email address
• Your nationality and visa status
• Your professional status and
• undergraduate students: please describe your background in set theory
• graduate students: please tell us your year and the name of your thesis advisor if you have one
• faculty: please tell us whether you hold a federal research grant
• A brief statement about your interest in the workshop

## Related activities

A few workshop participants who have arranged slightly longer visits to Toronto will speak on other days:
• Wednesday, May 27, in Bahen Centre 6183
• Gary Gruenhage (Auburn) at 13h30 on "Slim dense sets in products"
• Thursday, May 28, in Bahen Centre 6183
• Justin Moore (Cornell) at 13h30 on "Fast growth of the Folner function for Thompson's group F"
• Monday June 1, in Fields Institute 210
• KP Hart (Delft; Miami of Ohio) at 10h30
• Marion Scheepers (Boise) at 13h30
• Paul Larson (Miami of Ohio) at 15h30 on "Universally measurable sets in generic extensions"