David Fernandez: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number

Place: Fields Institute, Room 210
Date and time: Friday 23 January 2015 (13:30-15:00)
Speaker: David Fernandez
Title: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number.

Abstract: Strongly summable ultrafilters are a variety of ultrafilters that relate with Hindman’s finite sums theorem in a way that is somewhat analogous to that in which Ramsey ultrafilters relate to Ramsey’s theorem. It is known that the existence of these ultrafilters cannot be proved in ZFC, however such an existencial statement follows from having the covering of meagre to equal the continuum. Furthermore, using ultraLaver forcing in a short finite support iteration, it is possible to get models with strongly summable ultrafilters and a small covering of meagre, and these models will also have small dominating number. Using this ultraLaver forcing in a countable support iteration to get a model with small covering meagre and strongly summable ultrafilters is considerably harder, but it can be done and in this talk I will explain how (it involves a characterisation of a certain kind of strongly summable ultrafilter in terms of games). Interesingly, this way we also get the dominating number equal to the continuum, unlike the previously described model.

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