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## Connecticut Logic Seminar

- 10/28
*Connecticut Logic Seminar*

Closure algebras that are existentially closed

Philip Scowcroft (Wesleyan University)#### Connecticut Logic Seminar

Monday, October 28th, 2019

Closure algebras that are existentially closed

Philip Scowcroft (Wesleyan University)

4:45 PM - 6:00 PM

Storrs Campus

Exley Science Center 618, Wesleyan UniversityIn papers of 1944 and 1946, McKinsey and Tarski initiated the study of closure algebras—Boolean algebras equipped with an operation obeying a version of Kuratowski’s axioms—and in 1948 they applied their results to reach conclusions about intuitionistic logic and the modal logic S4. In 1982 Lipparini found non-elementary axioms for existentially closed (e.c.) closure algebras and showed that they do not form an elementary class. After outlining Lipparini’s results from a different perspective, this talk will provide new information about closure algebras that are e.c., (finitely or infinitely) generic, or algebraically closed.

Contact Information: Reed Solomon, david.solomon@uconn.edu More - 11/4
*Connecticut Logic Seminar*

Lowness of the Pigeonhole Principle

Benoit Monin (Creteil University)#### Connecticut Logic Seminar

Monday, November 4th, 2019

Lowness of the Pigeonhole Principle

Benoit Monin (Creteil University)

4:45 PM - 6:00 PM

Storrs Campus

MONT 214Given a coloring $c:\omega \rightarrow \{0, 1\}$, there must be, by the Pigeonhole principle, an infinite set $X$ such that $c$ assigns the same color to every element of $X$. This rather easy theorem is known as the $\mathrm{RT^1_2}$ principle in reverse mathematics : An instance of $\mathrm{RT^1_2}$ is a coloring $c:\omega \rightarrow \{0, 1\}$, and a solution of this instance is an infinite set $X$ whose every element are assigned the same color via $c$.

We study the general question of the computational power of $\mathrm{RT^1_2}$: Given a notion of computational strength, that is, an upward closed class $\mathcal{C}$ in the Turing degree, can we build an instance $c$ of $\mathrm{RT^1_2}$ such that every solution to $c$ is a member of $\mathcal{C}$? The general paradigm is that $\mathrm{RT^1_2}$ has very little computational power. For almost every known natural notion of computational strength $\mathcal{C}$, it is known that $\mathrm{RT^1_2}$ is low for this notion : for every instance $c$ of $\mathrm{RT^1_2}$, there is a solution of $c$ which is not a member of $\mathcal{C}$.

One of the first of these result is that for any non computable set $X$ and any instance $c$ of $\mathrm{RT^1_2}$, one solution of $c$ does not compute $X$ (Dzhafarov and Jockusch). To show this, the authors designed a special forcing notion : the computable Mathias forcing, with which one can control the truth of $\Sigma^0_1$ statements. Later, Monin and Patey designed a new forcing notion, that builds upon computable Mathias forcing, in order to control the truth of $\Sigma^0_2$ statements. This lead to the following result : for every instance $c$ of $\mathrm{RT^1_2}$, there is a solution of $c$ which is not of high degree, that is whose jump does not compute the double jump.

Later the same authors could obtain a generalization of this forcing in two ways : the first one to control the truth of $\Sigma^0_\alpha$ statements for $\alpha$ a computable ordinal, and the second one by "iterating" the forcing within product spaces, in order to obtain non-cohesive solutions, leading a separation of the reverse mathematical principles $\mathrm{RT^2_2}$ and $\mathrm{SRT^2_2}$.

Contact Information: Reed Solomon, david.solomon@uconn.edu More - 11/11
*Connecticut Logic Seminar*

Ramsey-Like Theorems and Moduli of Computation

Ludovic Patey (CNRS, Institut Camille Jordan, Lyon)#### Connecticut Logic Seminar

Monday, November 11th, 2019

Ramsey-Like Theorems and Moduli of Computation

Ludovic Patey (CNRS, Institut Camille Jordan, Lyon)

4:45 PM - 6:00 PM

Storrs Campus

MONT 214Ramsey's Theorem Asserts That Every $$k$$-Coloring of $$[\omega]^n$$ Admits an Infinite Monochromatic Set. On the other hand, for every computable $$k$$-coloring of $$[\omega]^2$$ and every non-computable set $$C$$, there is an infinite monochromatic set $$H$$ such that $$C \not \leq_T H$$. The latter property is known as \textit{cone avoidance}.

In this talk, we introduce a natural class of Ramsey-like theorems encompassing many problems studied in reverse mathematics. We show that this class admits a maximal statement satisfying cone avoidance and use it as criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every $$k$$-coloring of $$[\omega]^n$$, an infinite subdomain $$H \subseteq \omega$$ over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions.

Contact Information: Reed Solomon, david.solomon@uconn.edu More

*Past talks in or after Spring 2019 are accessible through the UConn Events Calendar.*

List of talks prior to Spring 2019.

List of talks prior to Spring 2019.