This Week in Logic at CUNY

Computational Logic Seminar
Tuesday, March 19, 2013 2:00 pm
Speaker: Stan Wainer The Leeds Logic Group, University of Leeds
Title: Computing Bounds from Arithmetical Proofs
We explore the role of the function a+2^x, and its generalizations to
higher number classes, in analyzing and measuring the computational
content of a broad spectrum of arithmetical theories. Newer
formalizations of such theories, based on the well-known normal/safe
variable separation of Bellantoni-Cook, enable uniform
proof-theoretical treatments of poly-time arithmetics, through Peano
arithmetic, and up to finitely iterated inductive definitions.

 

Models of PA
Wednesday, March 20, 2013 6:45 pm
Speaker: Stan Wainer The Leeds Logic Group, University of Leeds
Title: Fast Growing Functions and Arithmetical Independence Results
We explore the role of the function $a+2^x$ and its generalisations to
higher number classes, in supplying complexity bounds for the provably
computable functions across a broad spectrum of (arithmetically based)
theories. We show how the resulting “fast growing” subrecursive
hierarchy forges direct links between proof theory and various
combinatorial independence results – e.g. Goodstein’s Theorem (for
Peano Arithmetic) and Friedman’s Miniaturised Kruskal Theorem for
Labelled Trees (for $Pi^1_1$-CA$_0$).

Ref: Schwichtenberg and Wainer, “Proofs and Computations”, Persp. in
Logic, CUP 2012.

 

Set theory seminar
Friday, March 22, 2013 10:00 am
Speaker: Kaethe Minden The CUNY Graduate Center
Title: Ramsey ultrafilters
I will introduce the concept of a Ramsey ultrafilter and show that
under Martin’s Axiom, and under the continuum hypothesis, Ramsey
ultrafilters exists. I will actually show that this follows from some
consequences of MA on cardinal invariants of the continuum. If time
permits, I will make a connection to Ramsey’s theorem. This talk is
intended to bridge the gap between the previous talk by Miha Habic on
Martin’s Axiom and the upcoming talks by Victoria Gitman on Ramsey
cardinals.

CUNY Logic Workshop
Friday, March 22, 2013 2:00 pm
Speaker: Peter Koepke Rheinische Friedrich-Wilhelms-Universität Bonn
Title: Namba-like singularizations of successor cardinals
Bukowski-Namba forcing preserves aleph_1 and changes the cofinality of
aleph_2 to omega. We lift this to cardinals kappa > aleph_1 :
Assuming a measurable cardinal lambda we construct models over which
there is a further “Namba-like” forcing which preserves all cardinals
<= kappa and changes the cofinality of kappa^+ to omega. Cofinalities
different from omega can also be achieved by starting from measurable
cardinals of sufficiently strong Mitchell order. Using core model
theory one can show that the respective measurable cardinals are also
necessary. This is joint work with Dominik Adolf (Münster).

CUNY Logic Workshop
Friday, March 22, 2013 4:00 pm
Speaker: Philip Welch University of Bristol
Title: Determinacy in analysis and beyond
Recently Montalban and Shore derived precise limits to the amount of
determinacy provable in second order arithmetic. We review some of
the results in this area and recent work on lifting this to a setting
of ZF^- with a single measurable cardinal.

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