Rutgers Logic Seminar: Mondays

Information

Directions to the Hill Center can be found here. Please note that if you plan to drive, you will need a parking permit. This can be obtained from Hill 303, 305, or 307, though these offices close at 5:00 pm.

The

Seminar Schedule

5:00 pm - 6:00 pm, Room 705, Hill Center, Busch Campus

Upcoming Talks

Abstract: A well-known theorem of Shelah states that semiproperness of Namba forcing is equivalent to a version of Strong Chang's Conjecture. We consider a certain poset $\mathbb{Q}$ that: preserves stationary subsets of $\omega_1$ and kills the stationarity of the ground model's $[\omega_2]^\omega$ (like Namba forcing); yet preserves that $\omega_2$ has uncountable cofinality (unlike Namba forcing).

We prove that semiproperness of $\mathbb{Q}$ lies between two versions of Strong Chang's Conjecture (answering a question of Friedman-Krueger).

Past Talks

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.

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.

Abstract: We discuss some characterizations of large cardinals using model theory, especially around compactness in \mathbb{L}_{\kappa, \kappa}.

Abstract: 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.

This is joint work with John Goodrick and Byunghan Kim

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.

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 Sierpinskiâ€™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.

Title: Complete groups are complete co-analytic

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.

Title: Primitive Binary Structures

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.

Title: First-Order Logic with Isomorphism

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 and relevant slides can be found here.

Title: On Cichon's Diagram for Uncountable $\kappa$

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

Title: A Derived Model with a Measure

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.

Title: Unions of Chains of Signatures

Title: A Proof of Generation of Full Pointclasses

Previous Semesters

Fall 2016

Spring 2016