(This is joint work with Martin Pizarro). We prove that for any prime p the theory of separably closed fields of characteristic p is equational. This was known before for finite degree of imperfection.
In Ramsey Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple
In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.
In this talk, we will continue with basics of measurable cardinals and their relationship to non-trivial elementary embeddings. We proceed with basic facts about the constructible universe, L. After laying this groundwork, we show L cannot have a measurable cardinal. Time permitting, we will discuss the dichotomy introduced by Jensen's covering lemma: either L is a good approximation to V, or there is a non-trivial elementary embedding from L to L.
How complicated are countable torsion-free abelian groups? In particular, are they as complicated as countable graphs? In recent joint work with Shelah, we show it is consistent with ZFC that countable torsion-free abelian groups are $a \Delta^1_2$ complete; in other words, countable graphs can be encoded into them via an absolutely $\Delta^1_2$-map. I discuss this, and the related result: assuming large cardinals, it is independent of ZFC if there is an absolutely $\Delta^1_2$ reduction from Graphs to Colored Trees, which takes non-isomorphic graphs to non-biembeddable colored trees.
A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees.
In this talk, we will cover the basics of measurable cardinals and their relationship to non-trivial elementary embeddings. We proceed with basic facts about the constructible universe, L. After laying this groundwork, we show L cannot have a measurable cardinal. Time permitting, we will discuss the dichotomy introduced by Jensen's covering lemma: either L is a good approximation to V, or there is a non-trivial elementary embedding from L to L.
Forcing and elementary embeddings are central topics in set theory. Most of what set theorists have focused on are the study of forcing and elementary embeddings over models of ZFC. In this talk, we focus on forcing and elementary embeddings over models of the Axiom of Determinacy (AD). In particular, we focus on answering the following questions: work in V which models AD. Let P be a forcing poset and g ⊆ P be V -generic.
1) Does V [g] model AD?
2) Is there an elementary embedding from V to V [g]?
Regarding question 1, we want to classify what forcings preserve AD. We show that forcings that add Cohen reals, random reals, and many other well-known forcings do not preserve AD. Regarding question 2, an analogous statement to the famous Kunen’s theorem for models of ZFC, can be shown: suppose V = L(X) for some set X and V models AD, then there is no elementary embedding from V to itself. We conjecture that there are no elementary embeddings from V to itself. We present some of the results discussed above. There is still much work to do to completely answer questions 1 and 2. This is an ongoing joint work with D. Ikegami.
Shelah proved that a certain form of Strong Chang’s Conjecture is equivalent to the statement ``Namba forcing is semiproper". I will present some related results about semiproperness of ``nonreasonable” posets (a notion introduced by Foreman-Magidor). This is joint work with Hiroshi Sakai.
In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.