November 5: *Dorottya Sziraki*: Algebraic Logic and Vaught's Conjecture

16.00--17.00

Vaught's Conjecture states that if a countable first order theory has uncountably many pairwise non isomorphic countable models, then it has continuum many such models. In my talk, I will explain how we used methods from algebraic logic to prove a variant of Vaught's Conjecture. The variant is obtained by replacing, in the statement of the conjecture, the role of isomorphism with that of elementary embeddability, and by allowing only elementary embeddings that are in certain submonoids of injective functions. More precisely, for a countable theory T and a submonoid H of injective functions from \omega to \omega, let I(T,H) denote "the number" of pairwise non H-elementarily embeddable models of T with domain \omega. We prove that for \sigma-compact submonoids H, we have the following: if I(T,H) is uncountable, then I(T,H) is the continuum. Our proofs are based on the representation theory of cylindric algebras and topological properties of the Stone spaces of these algebras.

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November 19: *Kostas Tsaprounis*: Long Reals

16.00--17.00

Ever since Dedekind and Cantor, there is a well-known procedure which, starting with the set
of natural numbers, produces in a canonical fashion the complete ordered field of the reals. In
this talk, we study what happens when one replaces omega by any infinite cardinal kappa
in the above construction(s).
This is joint work in progress with David Aspero.

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December 5: *Volodya Shavrukov*: Astrology of Nerode semirings

16.00--17.00

Nerode semirings are models of the true forall-exists 1st order arithmetic that are finitely generated with respect to total recursive functions. They are aged at around half a century. They can also be seen as ultrapowers of N restricted to total recursive functions, with the suitable ultrafilter being one on the algebra of recursive sets.

We relate the structure of Nerode semirings to combinatorial amd complexity properties of recursive ultrafilters, and to how these ultrafilters are positioned w.r.t. various r.e. sets. We address issues such as ordering of skies in a Nerode semiring, existential completeness, and substructure lattices. We also discuss a couple of generalizations such as r.e. ultra- and prime powers.

This is joint work with James Schmerl.

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December 11: *Matthias Schirn*: Frege: Second-Order Abstraction and Referential Indeterminacy

16.00--18.00

In this talk, I shall critically discuss some issues related to Frege?s paradigms of second-order abstraction principles: Hume?s Principle and Axiom V. The focus is on the
referential indeterminacy of value-range terms arising from a semantic stipulation later to be embodied in Axiom V of Grundgesetze. I shall discuss Frege?s attempt to remove the
indeterminacy as well as his subsequent proof of referentiality for his formal language with special emphasis on the case of value-range terms. Attention will also be paid to the
assumptions that underly his overall strategy.

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December 12: *Jouko Väänänen*: Quantum Team Logic

16.00--17.00

A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy. We point out that these logical Bell's Inequalities are provable in the probability logic of Fagin, Halpern and Megiddo. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic first to probabilistic team semantics, and then to, what we call quantum team semantics.

This is joint work with G. Paolini and T. Hyttinen.

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November 19: *Kostas Tsaprounis*: Elementary embeddings and (very) large cardinals

16.00--17.00

In this (tutorial) talk we present some of the usual (very) large cardinals, which are those described by the existence of elementary embeddings between transitive class models of ZFC set theory. We look at some standard results and techniques in the context of such elementary embeddings. Subsequently, we introduce the hierarchies of C^(n)-cardinals, which were defined and studied by Bagaria, giving an overview of their properties and connections with the usual large cardinal hierarchy. Finally, we mention some recent applications of C^(n)-cardinals outside of set theory.

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June 30: *Benjamin Rin*: On Set-theoretic and Transfinite Analogues of Epistemic Arithmetic and Flagg Consistency

16.00--17.00

In the early ?80s, Shapiro proposed that we could understand and fruitfully analyze the pre-theoretic notion of computability within the framework of a modal version of arithmetic (essentially PA+S4) known as epistemic arithmetic (EA). Soon after, Flagg produced crucial evidence for Shapiro?s proposal in a well-known article that proved the consistency of Church?s Thesis with EA. (Here he used an epistemic version of Church?s Thesis, called ECT.) In this talk, I look at the possibility of extending Flagg?s result to a richer base theory, by replacing Peano arithmetic with set theory and, in turn, substituting ECT with an infinitary analogue?one that centers on Hamkins-Lewis infinite time Turing machines or even ordinal Turing machines in place of ordinary Turing machines. Doing this should clarify the limits of Flagg?s contribution, by informing us to what extent, if any, it depends on the built-in assumptions of finiteness inherent in traditional Turing computability.

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