Seminar talks spring 2020

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (1) the domain of interpretation is not empty (2) every name denotes exactly one object in the domain and (3) the quantifiers have existential import. Free logics usually reject (2), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names, namely self-identity, are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (1), are likewise briefly considered.

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These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization using sequent calculi, which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different systems.

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Final part of this paper is dedicated to extending the system with modalities by using a labelled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework.

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This presentation is part of joint work with Norbert Gratzl of MCMP, Munich.

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(% style="color: rgb(0,0,0);" %)Wed 4.3.2020 12-14, C124

Exam week

(% style="color: rgb(0,0,0);" %)Wed 11.3.2020 12-14, C124

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(% style="color: rgb(0, 0, 0); color: rgb(0, 0, 0); text-decoration: none" %)Nick Ramsey: (% style="color: rgb(0,0,0);" %)Kim-independence over arbitrary sets

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(% style="color: rgb(0,0,0);" %)Wed 18.3.2020 12-14, C124

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--Martin Lück: LTBA-- cancelled

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(% style="color: rgb(0,0,0);" %)Due to the corona virus situation, the seminar has moved online. Talks will primarily be 1 hour long.

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(% style="color: rgb(0,0,0);" %)Wed 25.3.2020 13.00-14, Zoom (%%)Meeting ID: 733 200 600, [[https:~~/~~/helsinki.zoom.us/j/733200600>>url:https://helsinki.zoom.us/j/733200600||rel="nofollow" shape="rect" class="external-link"]]

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(% style="color: rgb(0,0,0);" %)Jouko Väänänen: Fraenkel-Mostowski Models for Dependence Logic

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(% style="color: rgb(0,0,0);" %)Wed 1.4.2020 13.00-14, C124

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(%%)Joint Mathematical Physics and Mathematical Logic seminar, Zoom Meeting ID: 726 088 428, [[https:~~/~~/helsinki.zoom.us/j/726088428>>url:https://helsinki.zoom.us/j/726088428||rel="nofollow" shape="rect" class="external-link"]]

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MIP*=RE, Henry Yuen, University of Toronto

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What is the connection between Connes' embedding conjecture

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from the theory of von Neumann algebras, Tsirelson’s conjecture

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in quantum mechanics and theoretical computer science? Using the

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methods form the latter the former two conjectures were recently

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refuted.

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We will watch the recent seminar at IAS, Princeton by

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Henry Yuen with the help of local experts here at Helsinki.

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(% style="color: rgb(0,0,0);" %)Wed 8.4.2020 13-14, C124, Zoom (%%)Meeting ID: 321 760 683, [[https:~~/~~/helsinki.zoom.us/j/321760683>>url:https://helsinki.zoom.us/j/321760683||rel="nofollow" shape="rect" class="external-link"]]

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(% style="color: rgb(0,0,0);" %)Davide Quadrellaro: Algebraic Semantics for Propositional Dependence Logic

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(% style="color: rgb(0,0,0);" %)Wed 15.4.2020 13-14, C124

Easter, no seminar

(% style="color: rgb(0,0,0);" %)Wed 22.4.2020 13-14, Zoom (%%)Meeting ID: 695 5829 5917, [[https:~~/~~/helsinki.zoom.us/j/69558295917>>url:https://helsinki.zoom.us/j/69558295917||rel="nofollow" shape="rect" class="external-link"]]

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(% style="color: rgb(0,0,0);" %)Jose Iovino (San Antonio):(% style="color: rgb(0, 0, 0); text-decoration: none" %) Tao's Concept of Metastability as a Medium Connecting Disparate Areas of Mathematics

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(% style="color: rgb(0, 0, 0); text-decoration: none" %)Abstract: The concept of metastable convergence was introduced by Terry Tao as a tool for his 2008 ergodic theorem. It turns out that this concept is intimately related to ideas that had been used by logicians for decades. Moreover, it arises naturally in many other areas of mathematics, and it connects different subareas of logic in unexpected ways.

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(% style="color: rgb(0,0,0);" %)Wed 29.4.2020 13-14, Zoom Meeting ID: 634 1701 3230, (%%)[[https:~~/~~/helsinki.zoom.us/j/63417013230>>url:https://helsinki.zoom.us/j/63417013230||rel="nofollow" shape="rect" class="external-link"]]

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(% style="color: rgb(0,0,0);" %)Tuomas Hakoniemi: Feasible Interpolation for Algebraic Proof Systems

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(% style="color: rgb(0,0,0);" %)Wed 6.5.2020 12-14, C124

Exam week

(% style="color: rgb(0,0,0);" %)Wed 20.5.2020, Zoom, 16:00 local time (EEST). Link: [[https:~~/~~/us02web.zoom.us/j/83205906678>>url:https://us02web.zoom.us/j/83205906678||rel="nofollow" shape="rect" class="external-link"]], Meeting ID: 476 210 6037, password: HLGrp

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(% style="color: rgb(0,0,0);" %)John P. Burgess (Princeton): Measurable Selections: A Bridge Between Large Cardinals and Scientific Applications?

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(% style="color: rgb(0,0,0);" %)Abstract: There is no prospect of discovering measurable cardinals by radio astronomy, say by locating a pulsar pumping out the digits of zero-sharp, but this does not mean that higher set theory is entirely irrelevant to and unconnected with applied mathematics broadly construed. By way of example, the bearing of some celebrated descriptive-set-theoretic consequences of large cardinals on measurable selection theory, a body of results originating with a key lemma von Neumann’s work on the mathematical foundations of quantum theory, and further developed in connection with problems of mathematical economics, and one that perhaps deserves to be somewhat better known among logicians, will be considered from a philosophical point of view.

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(% style="color: rgb(0,0,0);" %)Handout [[here>>url:http://mathstat.helsinki.fi/logic/Burgess.pdf||rel="nofollow" shape="rect" class="external-link"]]. Video of the lecture [[here>>url:https://drive.google.com/open?id=1fe39M3W3rZtsH4zzwosWa7aAmN1fPXFu||rel="nofollow" shape="rect" class="external-link"]] (the very beginning is missing).

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Wed 27.5.2020, **15:00** local time, Zoom link: [[https:~~/~~/helsinki.zoom.us/j/66984743531?pwd=N2tNODhpdVBVZWg5eGVwQ0VDN3BwZz09>>url:https://helsinki.zoom.us/j/66984743531?pwd=N2tNODhpdVBVZWg5eGVwQ0VDN3BwZz09||rel="nofollow" shape="rect" class="external-link"]], Meeting ID: 669 8474 3531, Password: 429886

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Andrés Villaveces (Bogotá): One Puzzling Logic, Two Approximations and a Bonus

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**Abstract:** The puzzling logic (called (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L%5E1_%5Ckappa&bg=ffffff&fg=444444&s=0]](%%)for (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=%5Ckappa&bg=ffffff&fg=444444&s=0]](%%)a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L%5E1_%5Ckappa&bg=ffffff&fg=444444&s=0]](%%)has many properties that make it very well adapted to model theory, despite being stronger than(% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L_%7B%5Ckappa%2C%5Comega%7D&bg=ffffff&fg=444444&s=0]](%%). However, it also lacks a good syntactic definition.

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With Väänänen, we introduced the first approximation (called (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L%5E%7B1%2Cc%7D_%5Ckappa&bg=ffffff&fg=444444&s=0]](%%),) as a variant of (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L%5E1%5Ckappa&bg=ffffff&fg=444444&s=0]](%%)with a transparent syntax and many of the strong properties of Shelah’s logic.

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The second approximation (called Chain Logic), while not new (it is due to Karp), has been revisited recently by Dzamonja and Väänänen) also in relation to Shelah’s (% class="confluence-embedded-file-wrapper" %)[[image:url:https://s0.wp.com/latex.php?latex=L%5E1_%5Ckappa&bg=ffffff&fg=444444&s=0]](%%)and the Interpolation property.

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I will provide a description of these three logics, with emphasis on their relevance to model theory.

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As a bonus, I will make a connection between these logics and axiomatizing correctly an arbitrary AEC. This last part is joint work with Shelah.

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