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
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.
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.