The weekly announcements will be moved to the new website http://settheory.eu soon.
Vienna Research Seminar in Set Theory Time: Thursday, 9 October, 11:30-13:00 CEST Speaker: N. Chapman, TU Wien Title: Ranked Forcing and the Structure of Borel Hierarchies Abstract: The structural study of the Borel hierarchy on topological spaces is a foundational goal of descriptive set theory. By an early result of the field, we know that there exist universal sets at each level α<ω1 of the Borel hierarchy on the Baire space ωω, hence the order of this hierarchy, i.e. the first ordinal α at which every Borel set has been generated, attains …
The weekly announcements will be moved to the new website http://settheory.eu soon.
Vienna Research Seminar in Set Theory Time: Thursday, 9 October, 11:30-13:00 CEST Speaker: N. Chapman, TU Wien Title: Ranked Forcing and the Structure of Borel Hierarchies Abstract: The structural study of the Borel hierarchy on topological spaces is a foundational goal of descriptive set theory. By an early result of the field, we know that there exist universal sets at each level α<ω1 of the Borel hierarchy on the Baire space ωω, hence the order of this hierarchy, i.e. the first ordinal α at which every Borel set has been generated, attains the maximal possible value of ω1. However, there are other subspaces of ωω where this hierarchy is shorter; take for example any countable space, on which every Borel subset must be Σ02. The topic of this talk is a framework for the surgical alteration of the complexity of the Borel hierarchy on subspaces of ωω, pioneered by A. Miller. We will discuss Miller’s notion of α-forcing, which allows for either collapsing or increasing the length of the Borel hierarchy, as well as the proof ideas behind some preservation theorems necessary to do so. In the second part of the talk, we will delve into recent developments in this area, such as an extension of the framework to the field of generalized descriptive set theory of an uncountable cardinal κ=κ<κ or the study of the λ-Borel subsets of κκ for λ>κ, with a particular emphasis on the case λ=ω1 and κ=ω. We will give several examples of models constructed using this method in both the classical case of ω and the generalized case of an uncountable κ. Lastly, we will discuss some limitations of the technique and directions for future work. Information: This talk will be given in hybrid format. Please contact Petra Czarnecki for information how to participate.
Vienna Logic Colloquium Time: Thursday, 9 October, 15:00 – 15:50 CEST Speaker: R. Honzik, Charles U Prague Title: Compactness in mathematics Abstract: We discuss some well-known compactness principles for uncountable structures of small regular sizes ωn for 2≤n<ω, ℵω+1, ℵω2+1, etc.), consistent from weakly compact (the size-restricted versions) or strongly compact or supercompact cardinals (the unrestricted versions). For the exposition, we divide the principles into logical principles, which are related to cofinal branches in trees and more general structures (various tree properties), and mathematical principles, which directly postulate compactness for structures like groups, graphs, or topological spaces (for instance, countable chromatic and color compactness of graphs, compactness of abelian groups, Δ-reflection, Fodor-type reflection principle, and Rado’s Conjecture). We also focus on indestructibility, or preservation, of these principles in forcing extensions. While preservation adds a degree of robustness to such principles, it also limits their provable consequences. For example, some well-known mathematical problems such as Suslin Hypothesis, Whitehead’s Conjecture, Kaplansky’s Conjecture, and the categoricity of ω1-dense subsets of the reals (Baumgartner’s Axiom), are independent from some of the strongest forms of compactness at ω2. This is a refined version of Solovay’s theorem that large cardinals are preserved by small forcings and hence cannot decide many natural problems in mathematics. Additionally, we observe that Rado’s Conjecture plus 2ω=ω2 is consistent with the negative solutions, i.e. as they hold in V=L, of some of these conjectures (Suslin’s, Whitehead’s, and Baumgartner’s axiom), verifying that they hold in suitable Mitchell models. Finally, we comment on whether the compactness principles under discussion are good candidates for axioms. We consider their consequences and the existence or non-existence of convincing unifications (such as Martin’s Maximum or Rado’s Conjecture). This part is a modest follow-up to the articles by Foreman “Generic large cardinals: new axioms for mathematics?” and Feferman et al. “Does mathematics need new axioms?”. Information: This talk will be given in hybrid format. Please contact Petra Czarnecki for information how to participate.
New York Set Theory Seminar Time: Friday, 10 October, 11.00 New York time (17.00 CEST) Speaker: Dan Hathaway, University of Vermont Title: On Absoluteness Between V and HOD Abstract: We put together Woodin’s Σ21 basis theorem of AD+ and Vopěnka’s theorem to conclude the following: If there is a proper class of Woodin cardinals, then every (Σ21)uB statement that is true in V is true in HOD. Moreover, this is true even if we allow a certain parameter. We then show that stronger absoluteness cannot be implied by any large cardinal axiom consistent with the axiom V = Ultimate L. Information: Please see the seminar webpage.
New York Logic Workshop Time: Friday, 10 October, 14.00 New York time (20.00 CEST) Speaker: Philip Scowcroft, Wesleyan University Title: Injective simple dimension groups Abstract: A dimension group is a partially ordered Abelian group whose partial order is isolated and directed and has the Riesz interpolation property. A dimension group is simple just in case it has no nontrivial ideals, ideals being directed convex subgroups. By concentrating on the behavior of positive formulas in simple dimension groups, this talk will reveal a well-behaved part of their model theory. Information: Please see the seminar webpage.