Blog Archives

Topic Archive: maximality principle

Set theory seminarFriday, November 20, 201510:00 amGC 3212

Kaethe Minden

Subcomplete forcings

The CUNY Graduate Center

Subcomplete forcings are a class of forcings introduced by Jensen. These forcings do not add reals but may change cofinalities to $\omega$, unlike proper forcings. Examples of subcomplete forcings include Namba forcing, Prikry forcing, and any countably closed forcing. In this talk I will discuss some results concerning subcomplete forcing and the preservation of various properties of trees.

Set theory seminarFriday, March 20, 201510:00 amGC 6417

Philipp Lücke

Generalized Baire spaces and closed Maximality Principles

Rheinische Friedrich-Wilhelms-Universität Bonn

Given an uncountable regular cardinal $\kappa$, the generalized Baire space of $\kappa$ is set ${}^\kappa\kappa$ of all functions from $\kappa$ to $\kappa$ equipped with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$.
A subset of this space is $\mathbf{\Sigma}^1_1$ (i.e. a projection of a closed subset of ${}^\kappa\kappa\times{}^\kappa\kappa$) if and only it is definable over $\mathrm{H}(\kappa^+)$ by a $\Sigma_1$-formula with parameters. This shows that the class of $\mathbf{\Sigma}^1_1$-subsets contains a great variety of set-theoretically interesting objects. Moreover, it is known that many basic and interesting questions about sets in this class are not decided by the axioms of $\mathrm{ZFC}$ plus large cardinal axioms.

In my talk, I want to present examples of extensions of $\mathrm{ZFC}$ that settle many of these questions by providing a nice structure theory for the class of $\mathbf{\Sigma}^1_1$-subsets of ${}^\kappa\kappa$. These forcing axioms appear in the work of Fuchs, Leibman, Stavi and Väänänen. They are variations of the maximality principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.

George Leibman
Bronx Community College, CUNY
George Leibman is a professor in the Mathematics and Computer Science department at Bronx Community College, CUNY. He received his doctorate from the CUNY Graduate Center in 2004, under the direction of Joel Hamkins, and he conducts research in set theory, with a particular interest in the modal logic of forcing.
Joel David Hamkins
The City University of New York
Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite.  He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research.  He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess.  His work on the automorphism tower problem lies at the intersection of group theory and set theory.  Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.