Between Sacks and random for inaccessible kappa
Mittwoch, 22.10.14, 16:30-17:30, Raum 404, Eckerstr. 1
A non-trivial issue concerning tree-like forcings in the generalized framework is to introduce a random-like forcing, where random-like means to be κκ-bounding, < κ-closed and κ+-cc simultaneously. Shelah managed to do that for κ weakly compact. In this talk we aim at introducing a forcing satisfying these three properties for κ inaccessible, and not necessarily weakly compact. This is joint work with Sy Friedman.
Reflection principles
Mittwoch, 29.10.14, 16:30-17:30, Raum 404, Eckerstr. 1
Basic Notions: Forcing
Mittwoch, 5.11.14, 16:30-17:30, Raum 404, Eckerstr. 1
Y-c.c. and Y-proper forcings
Mittwoch, 12.11.14, 16:30-17:30, Raum 404, Eckerstr. 1
Representing sets of cofinal branches as continuous images
Mittwoch, 19.11.14, 16:30-17:30, Raum 404, Eckerstr. 1
Let \(\bkappa\) be an infinite cardinal and \(T\) be a tree of height \(\bkappa\). We equip the set \([T]\) of all branches of length \(\bkappa\) through \(T\) with the topology whose basic open subsets are sets of all branches containing a given node in \(T\). Given a cardinal \(\bnu\), we consider the question whether \([T]\) is equal to a continuous image of the tree of all functions \(s:\balpha\blongrightarrow\bnu\) with \(\balpha<\bkappa\). This is joint work with Philipp Schlicht.\n\n
Overview of the generalized combinatorial cardinal characteristics
Mittwoch, 26.11.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will talk about how the combinatorial cardinal characteristics\nreviewed in [Blass, Combinatorial Cardinal Characteristics of the\nContinuum] can be generalized to uncountable cardinals kappa and what is\nknown about consistency results for them.\n\n
Silver trees and Cohen reals
Mittwoch, 3.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will sketch the main ideas of my recent result that the\nmeager ideal is Tukey reducible to the Mycielski ideal. The latter one is\nthe ideal associated with Silver forcing. This implies that every\nreasonable amoeba forcing for Silver adds a Cohen real. This has been open\nfor some years.\n
Schanuel's Conjecture and Exponential Fields
Mittwoch, 10.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
Schanuel's Conjecture states that for a collection of n complex\nnumbers z1, ..., zn, linearly independent over the field of\nrational numbers, the transcendence degree of z1, ..., zn,\nexp(z1), ..., exp(zn) is at least n.\n\nZilber constructs in [Zilber, Pseudo-exponentiation on\nalgebraically closed fields of characteristic zero] a sentence\nwhose models are structures called strongly\nexponentially-algebraically closed fields with\npseudo-exponentiation, which are unique in every uncountable\ncardinality. One of their main properties is that Schanuel's\nConjecture holds in those fields.\n\nFirstly, I will outline the properties of Zilber's fields.\nSecondly, I will sketch the proof given in [Marker,\nA Remark on Zilber's Pseudoexponentiation] showing that, if one\nassumes Schanuel's Conjecture, the simplest case of one of the\naxioms of Zilber's fields holds in the complex exponential field.\n
Amoeba and tree ideals
Mittwoch, 17.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will talk about what I asked to Spinas in the end of his talk, i.e., whether an\namoeba for Silver might add Cohen reals. Two weeks ago he proved that add(J(Silver)) is at most\nadd(M). However this is not strictly sufficient to infer that any proper amoeba for Silver does\nadd Cohen reals, but only that it does not have the Laver property. I will clarify this\nissue. If there will be any time left I will also present some results about other tree ideals,\nwhich are part of a joint work, still in preparation, with Yurii Khomskii and Wolfgang\nWohofsky.\n
Cardinal characteristics at supercompact kappa in the small u(kappa), large 2^kappa model
Mittwoch, 14.1.15, 16:30-17:30, Raum 404, Eckerstr. 1
When generalising arguments about cardinal characteristics of the continuum to cardinals kappa greater than omega, one frequently comes up against the problem of how to ensure that a filter built up through an iterated forcing remains kappa complete at limit stages of small cofinality. A technique of Dzamonja and Shelah is useful for overcoming this problem; in particular, there is a natural application of this technique to obtain a model in which 2^kappa is large but the ultrafilter number u(kappa) is kappa^+. After introducing this model, I will talk about joint work with Vera Fischer (Technical University of Vienna) and Diana Montoya (University of Vienna) calculating many other cardinal characteristics at kappa in the model and its variants.
On the theory of universal specializations of Zariski structures
Mittwoch, 21.1.15, 16:30-17:30, Raum 404, Eckerstr. 1
Partially definable forcing and weak arithmetics
Mittwoch, 4.2.15, 16:30-17:30, Raum 404, Eckerstr. 1
Given a nonstandard model M of arithmetic we want to expand it\nby interpreting a binary relation symbol R such that R^M does something\nprohibitive, e.g. violates the pigeonhole principle in the sense that R^M\nis a bijection from n+1 onto n for some (nonstandard) n in M. The goal is\nto do so saving as much as possible from ordinary arithmetic. More\nprecisely, we want the expansion to satisfy the least number principle for\na class of formulas as large as possible. We describe a forcing method to\nproduce such expansions and revisit the most important results in the\narea.\n
A new forcing order
Mittwoch, 11.2.15, 16:30-17:30, Raum 404, Eckerstr. 1