Topic Archive: Boolean algebras
Set theory seminarFriday, September 18, 201510:00 amGC 3212
The City University of New York
I will present a criterion for when an ultrafilter on a Boolean algebra gives rise to the Bukovsky-Dehornoy phenomenon, namely that the intersection of all intermediate ultrapowers is equal to the the Boolean model. Time permitting, I will show that the Boolean algebras of Prikry and Magidor forcing satisfy the strong Prikry property, and that these forcings come with a canonical imitation iteration whose limit model is the Boolean ultrapower by a very canonical ultrafilter on their respective Boolean algebras.
Set theory seminarFriday, April 24, 201510:00 amGC 6417
CUNY Borough of Manhattan Community College
The Boolean ultrapower construction is a natural generalization of the classical ultrapower construction, but the Boolean ultrapower uses an ultrafilter on a complete Boolean algebra instead of a set. It was initially unknown as to whether in ZFC there exists a Boolean ultrapower which is not always isomorphic to a classical ultrapower. This problem was resolved in 1976 by Bernd and Sabine Koppelberg who constructed a Boolean ultrapower which is not an ultrapower. On the other hand, there does not seem to be any reference in the mathematical literature to atomless Boolean ultrapowers which are isomorphic to classical ultrapowers.
We shall first generalize the notion of a Boolean ultrapower to the notion of a BPA-ultrapower which is in a sense the most general ultrapower construction. Then by applying a result of Joel David Hamkins which characterizes the Boolean ultrapowers which are classical ultrapowers, we shall investigate examples of Boolean ultrapowers which are not classical ultrapowers as well as Boolean ultrapowers which are classical ultrapowers. For instance, I claim that under GCH every complete atomless Boolean algebra has an ultrafilter which gives rise to a Boolean ultrapower which is not a classical ultrapower. On the other hand, using the Keisler-Shelah isomorphism theorem, we may construct Boolean ultrapowers in ZFC on a fairly general class of Boolean algebras which are classical ultrapowers.
Models of PAMonday, March 10, 20146:30 pmGC 4214.03
The City University of New York
In his 1976 paper Haim Gaifman proved that for every set I, every model M of PA has an elementary end extension N such that Lt(N/M) is isomorphic to P(I). I will present a proof.
University of Denver
Professor Dobrinen earned her Ph.D. at the University of Minnesota under Karel Prikry in 1996, afterwards holding post-doctoral positions at Penn State and the University of Vienna before moving to the University of Denver. Her research interests mainly fall under the broad category of logic and foundations of Mathematics, and includes research in set theory, Ramsey theory, Boolean algebras, and measure theory. She has investigated relationships between random reals, eventually dominating functions, measure, generalized weak distributive laws, infinitary two-player games, and complete embeddings of the Cohen algebra into complete Boolean algebras. Currently, she is working on problems in Ramsey theory, problems regarding the structure of the Tukey types of ultrafilters, and problems involving both.
Professor Gaifman’s first result (obtained when he was a math student) was the equivalence of context-free grammars and categorial grammars. He was Carnap’s research assistant, working on the foundations of probability theory, and got his Ph. D. under Tarski (on infinite Boolean algebras). He worked on a broad spectrum of subjects: in mathematical logic (mostly set theory, where he invented the technique of iterated ultrapowers, and models of Peano’s arithmetic), foundations of probability (where he defined probabilities on first-order and on richer languages), in philosophy of language and philosophy of mathematics, as well as in theoretical computer science.. He held various permanent and visiting positions in mathematics, philosophy and computer science departments. While he was professor of mathematics at the Hebrew University, he taught courses in philosophy and directed the program in History and Philosophy of Science.
Gaifman’s recent interests include foundations of probability, rational choice, philosophy of mathematics, logical systems that formalize aspects of natural reasoning, Frege and theories of naming.
CUNY Logic WorkshopFriday, September 14, 201212:00 amGC 6417
City University of New York
An abstract of this talk will be added.