This Week in Logic at CUNY:

– – – – Monday, Oct 24, 2011 – – – –

– – – – Tuesday, Oct 25, 2011 – – – –

Computational Logic Seminar

October 25. Room 3309. Time 2:00 – 4:00 PM

Speaker: Junhua Yu (Graduate Center)

Title: Self-referentiality of the Brouwer-Heyting-Kolmogorov semantics for intuitionistic logic.

Abstract. The problem of formalizing the intended provability semantics for intuitionistic propositional logic IPC

(Brouwer-Heyting-Kolmogorov semantics, BHK semantics for short) was solved within Goedel-Artemov’s framework. According to this approach, IPC is embedded into modal logic S4 by Goedel translation, S4 is embedded into the Logic of Proofs LP via Artemov’s realization, and LP has a natural provability interpretation in Peano arithmetic. A principal feature of Artemov’s realization of S4 is that it makes use of self-referential LP-formulas of the form c:A(c), i.e., ‘c is a proof of a formula containing c itself.’ Kuznets showed that there are S4-theorems that are impossible to be realized without using self-referential LP-formulas. In this paper we adapt Kuznets’ method to find IPC-theorems that call for such direct self-referentiality in LP. This suggests that BHK semantics of IPC is intrinsically self-referential.

– – – – Wednesday, Oct 26, 2011 – – – –

– – – – Thursday, Oct 27, 2011 – – – –

– – – – Friday, Oct 28, 2011 – – – –

Set Theory Seminar

Friday, October 28, 2011 10:00 am GC 6417

Ms. Erin K. Carmody (Ph.D. Program in Mathematics, Graduate Center of CUNY) Prikry forcing, generic sets, and iterated ultrapowers

Abstract. First I’ll introduce Prikry forcing which preserves cardinals and adds a cofinal omega sequence to a measurable cardinal kappa. I will prove the Mathias characterization which explains when a cofinal omega sequence is Prikry generic. Let U be a normal measure on kappa and let V–>M_0–>M_1–>… ->M_omega be the system of embeddings obtained from iterating U. I’ll prove a theorem of Solovay which says that the sequence of critical points s= obtained from iterating a measure on kappa, omega many times, is M_omega-generic for Prikry forcing on j_omega(kappa). I’ll also show that this generic extension M_omega[s] is equal to the intersection of the M_n.

Model Theory Seminar

Friday, October 28, 2011 12:30 pm GC 6417

Mr. Thomas Ferguson (Ph.D. Program in Philosophy, Graduate Center of CUNY) Definability of indiscernible types of PA

Logic Workshop

Friday, October 28, 2011 2:00 pm GC 6417

Professor Benedikt Löwe (Universiteit van Amsterdam)

Multiplication in the hierarchy of norms

Abstract. A norm is a surjective function from the reals onto an ordinal. Given two norms φ and ψ, we can define a Wadge-like game G(φ ψ): if player I plays x and player II plays y, player II wins if and only if φ(x) ≤ ψ(y). If player II has a winning strategy in G(φ ψ), we write φ ≤ ψ . The equivalence classes of the derived equivalence relation form the hierarchy of norms.

Under the assumption of the axiom of determinacy (AD), the hierarchy of norms is a wellordering. In [1], Duparc investigated the Borel fragment of the hierarchy; in [2], we investigated the full hierarchy under the assumption of AD, and gave a lower bound of Θ2 for its length (where Θ := sup {α : there is a surjection from R onto α}). In this talk, we investigate further closure properties of the hierarchy of norms, in particular, the existence of an analogue of the multiplication for sets in the Wadge hierarchy, i.e., an operation such that (for well-chosen φ and ψ) the ordinal rank of φ⊗ψ is at least as big as the product of the ordinal ranks of φ and ψ. This closure property allows us to improve the lower bound to Θω.

(Bibliography: http://nylogic.org/Workshop/Fall2011/BenediktLoewe)

Seminar in Logic and Games

Friday, October 28, 4:15 PM

CUNY Graduate Center, room 4419

Jan Plaza (SUNY Plattsburgh)

Representation of Strict Closure Space Algebras

Abstract. Strict closure spaces are a generalization of topological spaces in which the intersection of two open sets does not need to be open. We define strict closure space algebras as Boolean algebras augmented with a unary operation Int that satisfies conditions analogous to those of the interior operation from strict closure spaces. We also consider a first order modal logic SCS in which the necessity operator satisfies analogous conditions. Sikorski proved that every interior algebra can be represented as an algebra of subsets of a zero-dimensional topological space that is a subspace of a Cantor cube 2^n; in this paper we prove an analogous result for strict closure space algebras. We also prove representability of Tarski-Lindenbaum algebras of SCS-theories in first-order languages of arbitrary cardinality.

Keywords: strict closure spaces, topology, Cantor cube, interior algebras, topological Boolean algebras, representation theorem, preserving infinite joins and meets.

Next Week in Logic at CUNY:

– – – – Monday, Oct 31, 2011 – – – –

– – – – Tuesday, Nov 1, 2011 – – – –

– – – – Wednesday, Nov 2, 2011 – – – –

– – – – Thursday, Nov 3, 2011 – – – –

– – – – Friday, Nov 4, 2011 – – – –

Set Theory Seminar

Friday, November 4, 2011 10:00 am GC 6417

Mr. Spencer Unger (Carnegie Mellon University)

New Branch Lemmas

Abstract. A branch lemma is a statement of the form “Forcing of type X cannot add a branch through a tree of type Y.” Branch lemmas often form an essential part of arguments involving forcing and large cardinals. As a warm up I’ll state and prove two classical branch lemmas. As a sample application, I will give a brief sketch of a forcing argument due to Mitchell for obtaining the tree property at ω2 from a weakly compact cardinal. I will then state a generalization of each classical branch lemma. I will prove one of the lemmas and mention the technique of the proof of the other. To conclude I will mention applications of both lemmas and time permitting I will sketch the proof of one the applications.

Model Theory Seminar

Friday, November 4, 2011 12:30 pm GC 6417

Speaker TBA

More on finitely generated models of PA

Logic Workshop

Friday, November 4, 2011 2:00 pm GC 6417

Dr. Samuel Coskey (The Fields Institute and York University) The Borel Tukey order on cardinal invariants

Abstract. Many cardinal invariants admit a natural definition of the form min { |F| : ∀x ∃y∈F xRy }

where R is some relation, x ranges over the domain of R, and F ranges over subsets of the codomain of R. For instance, the dominating number is defined by the relation ≤* (domination mod finite) on ωω.

If R and R’ are two relations, then R is said to be above R’ in the Tukey order iff there exist maps φ from the domain of R’ to the domain of R and ψ from the codomain of R to the codomain of R’ such that φ(x) R y ⇒ x R’ ψ(y)

The Tukey ordering is important because it corresponds very closely with inequality of the associated cardinal invariants. However, in practice one is more concerned with true inequalities, that is, inequalities which hold in all models of ZFC. For this reason, Blass proposed that we consider the Borel Tukey order, which is defined as above except that now R,R’ are assumed to be on standard Borel spaces and the maps φ and ψ are required to be Borel. The Borel Tukey order is known to have applications of a combinatorial nature in areas such as parameterized diamond principles and Borel equivalence relations.

In this talk, we will build upon some work of Mildenberger on the Borel Tukey ordering for a family of unsplitting relations. More generally, we will discuss the similarities and differences between the usual ordering and the Borel Tukey orderings on a modest collection of classical combinatorial cardinal invariants. For this, we will need to widen our attention slightly to cardinals which admit a definition of the form

min { |F| : P(F) & ∀x ∃y∈F xRy }

where R is as above and P is some second order property of the families F. This will allow us to consider the Borel Tukey order on many more cardinals. For instance, we shall be able to speak of the pseudo-intersection number, which is defined to be the least cardinality of a family F such that F is centered and ∀x∈[ω]ω ∃y∈F such that x⊄*y.

This is joint work with Juris Steprāns and Tamás Mátrai.

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