Title: Logic with Probability Quantifiers

Abstract: This talk is based on chapter XIV of Model-Theoretic Logics
(https://projecteuclid.org/euclid.pl/1235417263#toc). I will first give
a brief review of admissible sets and the infinitary logic which is
necessary for probability quantifiers. Then I will present the language
of probability quantifiers, as well as the proof theory, model theory,
and some examples which indicate the expressive power of the language.
Time permitting, my goal is to work towards the main completeness
theorem in section 2.3

Title: Amalgamating generic reals, a surgical approach
Location: Keller Hall 313
Speaker: Kameryn Williams, UHM

The material in this talk is an adaptation of joint work with Miha Habič, Joel David Hamkins, Lukas Daniel Klausner, and Jonathan Verner, transforming set theoretic results into a computability theoretic context.

Let $\mathcal D$ be the collection of dense subsets of the full binary tree coming from a fixed countable Turing ideal. In this talk we are interested in properties of $\mathcal D$-generic reals, those reals $x$ so that every $D \in \mathcal D$ is met by an initial segment of $x$. To be more specific the main question is the following. Fix a real $z$ which cannot be computed by any $\mathcal D$-generic. Can we craft a family of $\mathcal D$-generic reals so that we have precise control over which subfamilies of generic reals together compute $z$?

I will illustrate a specific of this phenomenon as a warm up. I will show that given any $\mathcal D$-generic $x$ there is another $\mathcal D$-generic $y$ so that $x \oplus y$ can compute $z$. That is, neither $x$ nor $y$ can compute $z$ on their own, but together they can.

The main result for the talk then gives a uniform affirmative answer for finite families. Namely, I will show that for any finite set $I = \{0, \ldots, n-1\}$ there are mutual $\mathcal D$-generic reals $x_0, \ldots, x_{n-1}$ which can be surgically modified to witness any desired pattern for computing $z$. More formally, there is a real $y$ so that given any $\mathcal A \subseteq \mathcal P(I)$ which is closed under superset and contains no singletons, that there is a single real $w_\mathcal{A}$ so that the family of grafts $x_k \wr_y w_\mathcal{A}$ for $k \in A \subseteq I$ can compute $z$ if and only if $A \in \mathcal A$. Here, $x \wr_y w$ is a surgical modification of $x$, using $y$ to guide where to replace bits from $x$ with those from $w$.