Mathematics Department - Logic Seminar - Spring 2017

# Logic Seminar - Spring 2017

### Organizer(s)

Grigor Sargsyan, Rebecca L Coulson

### Website

http://www.math.rutgers.edu/~rlg131/logicseminar.html

## Monday, April 24th

### "Simultaneous stationary reflection and failure of SCH "

Time: 5:00 PM
Location: Hill 705
Abstract: We will show that it is consistent to have finite simultaneous stationary reflection at $kappa^+$ with not SCH at $kappa$. This extends a result of Assaf Sharon. We will also present an abstract approach of iterating Prikry type forcing and use it to bring our construction down to $aleph_omega$.

This is joint work with Assaf Rinot.

## Monday, April 17th

### " ISP and SCH"

Time: 5:00 PM
Location: Hill 705
Abstract: ISP($kappa$) is a tree property-like principle, introduced by Weiss to capture the combinatorial essence of supercompactness. For $kappa$ inaccessible, ISP($kappa$) holds if and only if $kappa$ is supercompact. However, it is consistent relative to a supercompact that ISP holds at accessible cardinals (e.g. $aleph_2$) and such instances still entail some of the same consequences as supercompactness. For example, in analogy with Solovay's theorem that SCH holds above a supercompact cardinal, there is Viale's result that SCH holds above $kappa$ assuming ISP($kappa$) plus the existence of enough internally unbounded structures. Does ISP($kappa$) alone imply SCH above $kappa$? This would follow from a positive answer to a question of Viale and Weiss: Are all $aleph_1$-guessing models internally unbounded? We give partial negative answers to both questions, using ideas of Sinapova and Unger.

## Monday, April 10th

### "Model-theoretic characterizations of large cardinals "

Time: 5:00 PM
Location: Hill 705
Abstract: We discuss some characterizations of large cardinals using model theory, especially around compactness in mathbb{L}_{kappa, kappa}.

## Monday, April 3rd

### "Homology groups for types in stable theories"

Time: 5:00 PM
Location: Hill 705
Abstract: (Joint work with John Goodrick and Byunghan Kim.)

This talk will discuss the study of the type amalgamation properties in first-order theories by means of certain homology groups of types. The main focus of the talk will be on the theorem saying that if a first-order theory T is stable and n is the smallest natural number such that the n-th homology group of a strong type p is non-trivial, then the n-th homology group of p is isomorphic to the automorphism group of a specific part of the algebraic closure of n independent realizations of p. A by-product of the analysis is the conclusion that the automorphism group must be abelian.

## Monday, March 27th

### "Compactness of $omega_1$"

Time: 5:00 PM
Location: Hill 705
Abstract: We investigate various aspects of compactness of omega_1 under ZF + DC (the Axiom of Dependent Choice). We say that omega_1 is X-supercompact if there is a normal, fine, countably complete nonprincipal measure on powerset_{omega_1}(X) (in the sense of Solovay). We say omega_1 is X-strongly compact if there is a fine, countably complete nonprincipal measure on powerset_{omega_1}(X). A long-standing open question in set theory asks whether (under ZFC) "supercompactness" can be equiconsistent with "strong compactness. We ask the same question under ZF+DC. More specifically, we discuss whether the theories "omega_1 is X-supercompact" and "omega_1 is X-strongly compact" can be equiconsistent for various X. The global question is still open but we show that the local version of the question is false for various X. We also discuss various results in constructing and analyzing canonical models of AD^+ + omega_1 is X-supercompact.

## Monday, March 20th

### "The cube problem for linear orders"

Time: 5:00 PM
Location: Hill 705
Abstract: In the 1950s, Sierpinski asked whether there exists a linear order that is isomorphic to its lexicographically ordered cartesian cube but not to its square. The analogous question has been answered positively for many different classes of structures, including groups, Boolean algebras, topological spaces, graphs, partial orders, and Banach spaces. However, the answer to Sierpinski’s question turns out to be negative: any linear order that is isomorphic to its cube is already isomorphic to its square, and thus to all of its finite powers. I will present an outline of the proof and give some related results.

## Monday, March 6th

### "Complete groups are complete co-analytic"

Time: 5:00 PM
Location: Hill 705
Abstract: A group G is said to be complete if G is centerless and every automorphism of G is inner. In this talk, answering a question of Kechris, I will show that the set of countably infinite complete groups is complete co-analytic in the Polish space of countably infinite groups.

## Monday, February 27th

### " Primitive Binary Structures"

Time: 5:00 PM
Location: Hill 705
Abstract: I discuss the theory of relational complexity of finite structures, and two types of open problem: the computation of relational complexity in natural cases and the determination of the infinite families of finite primitive structures having bounded relational complexity. The first is a problem in combinatorics and the second is a problem in permutation group theory. Important progress on the second has been made recently by Gill and Spiga.

## Monday, February 20th

### " First-Order Logic with Isomorphism"

Time: 5:00 PM
Location: Hill 705
Abstract: We describe an extension of the syntax and proof system of first-order logic that has a natural semantics in the Univalent Foundations. This allows us to carry out a model theory in which mathematical structures are formalized in terms of homotopy types, just as in traditional model theory they are formalized in terms of sets. After defining the system, we will outline the relevant soundness and completeness results and sketch some applications.

This talk is based on the paper here: (https://arxiv.org/abs/1603.03092) and relevant slides can be found here: (http://rci.rutgers.edu/~dt506/HMT.online.version.pdf).

## Monday, February 13th

### "On Cichon's Diagram for Uncountable $kappa$"

Time: 5:00 PM
Location: Hill 705
Abstract: Cardinal invariants of the Baire space $omega^{omega}$ have been widely studied and understood. In this talk I will mention our work aiming to study the cardinal invariants of Cichon's Diagram when considering its generalization to the generalized Baire space $kappa^{kappa}$, where $kappa$ is an uncountable cardinal. Our research focuses mainly on the cardinals in the diagram associated with the $kappa$-Meager ideal, due to the absence of a notion of measure on these spaces. I will present the results that can be easily lifted from the countable case as well as some differences and open problems that arise when trying to achieve such a generalization.

This is joint work with Jorg Brendle, Andrew Brooke-Taylor, and Sy-David Friedman

## Monday, February 6th

### "A derived model with a measure"

Time: 5:00 PM
Location: Hill 705
Abstract: By a slight modification of Steel's stationary-tower-free proof of the derived model theorem, I will give an outline of how to get a canonical model of AD^+ with a measurable cardinal above Theta, assuming a limit of Woodin cardinals with a measurable above.

## Monday, January 30th

### "Unions of Chains of Signatures"

Time: 5:00 PM
Location: Hill 705

## Monday, January 23rd

### "A proof of Generation of Full Pointclasses"

Time: 5:00 PM
Location: Hill 705

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