This is the sixth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We will begin discussing the Borel hierarchy.
In first order logic, the Baldwin-Lachlan characterization of $\aleph_1$-categorical
theories implies that the notion is absolute between transitive models of set theory.
Here, we seek a similar characterization for having a unique atomic model of size $\aleph_1$.
At present, we have several conditions that imply many non-isomorphic atomic models of size $\aleph_1$.
Curiously, even though the results are in ZFC, their proofs rely on forcing.
This is joint work with John Baldwin and Saharon Shelah.
This is the fifth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.
This is the fourth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.
This is the third in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.
Recently, nonstandard and ultrafilter methods have been used to obtain a number of significant results in Combinatorial Number Theory. In this talk I will provide a brief overview of some recent work in this area, focusing on the use of nonstandard methods in problems involving the existence of various types of structured sets contained in subsets of the natural numbers that satisfy various density conditions.
This is the second in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. The topics discussed will include tree representations, universality properties of Polish spaces, and subspaces of Polish spaces.
This is the first in a series of introductory lectures in descriptive set theory, following Matt Foreman's expository paper. The topics discussed will be basics of Polish topologies, product topologies, Cantor space and Baire space, and infinite trees.
Tree properties are a family of combinatorial principles that characterize large cardinal properties for inaccessibles, but can consistently hold for "small" (successor) cardinals such as $\aleph_2$. It is a classic theorem of Magidor and Shelah that if $\kappa$ is the singular limit of supercompact cardinals, then $\kappa^+$ has the tree property. Neeman showed how to force $\kappa^+$ to become $\aleph_{\omega+1}$ while maintaining the tree property. Fontanella generalized these results to the strong tree property.
We show (in ZFC) that if $\kappa$ is a singular limit of supercompact cardinals, then $\kappa^+$ has the super tree property (this jump from "strong" to "super" is analogous to the jump in strength from strongly to supercompact cardinals). We remark on how to get the super tree property at $\aleph_{\omega+1}$, and on some interesting consequences for the existence of guessing models at successors of singulars. This is joint work with Dima Sinapova.
In this continuation of my talk from last week, I will introduce the notion of a spectral gap subalgebra of a tracial von Neumann algebra and show how it connects to the definability of relative commutants. I will also mention some applications of these results. I will introduce all notions needed from the theory of von Neumann algebras.