**Computational Logic Seminar**

** September 4, Time 2:00 – 4:00 PM, Room 3309.**

Speaker: Tudor Protopopescu (Graduate Center)

Title: Discovering Knowability: A Semantic Analysis (a joint work with S.

Artemov, just appeared in Synthese)

Abstract: We provide a semantic analysis of the well-known knowability

paradox stemming from the Church-Fitch observation that the meaningful

knowability principle all truths are knowable, when expressed as a bi-modal

principle

F -> <>KF,

yields an unacceptable omniscience property all truths are known. We offer

an alternative semantic proof of this fact independent of the Church-Fitch

argument. This shows that the knowability paradox is not intrinsically

related to the Church-Fitch proof, nor to the Moore sentence upon which it

relies, but rather to the knowability principle itself. Further, we show

that, from a verifiability perspective, the knowability principle fails in

the classical logic setting because it is missing the explicit

incorporation of a hidden assumption of stability: `the proposition in

question does not change from true to false in the process of discovery.’

Once stability is taken into account, the resulting stable knowability

principle and its nuanced versions more accurately represent

verification-based knowability and do not yield omniscience.

**Set Theory Seminar**

** Friday, Sep 07, 2012, 2:00pm, GC 6417**

Norman Perlmutter (The Graduate Center of the City University of New York)

Woodin for Supercompactness and Vopenka cardinals

Abstract. I present a tentative result that Woodin for supercompactness

cardinals are equivalent to Vopenka cardinals. This result is vaguely

hinted at, though not proven, in Kanamori’s text, and I believe I have

worked out the details. Kappa is Vopenka iff for every collection of kappa

many model-theoretic structures with domain subset of V_kappa, there exists

an elementary embedding between two of them. Kappa is Woodin for

supercompactness if it meets the definition of a Woodin cardinal, with

strongness replaced by supercompactness. That is to say, for every function

f:kappa to kappa, there exists a closure point delta of f and an elementary

embedding j:V –>M such that j(delta) j(f)(delta) sequences.

**Logic Workshop**

** Friday, Sep 07, 2012 GC 6417**

Professor Joel David Hamkins (The City University of New York)

Recent progress on the modal logic of forcing and grounds

Abstract. The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and “true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.