Colloquium: Tuesday Jan. 7, 2025. Speaker: Yair Hayut (Hebrew University). Title: “Dense Ideals”.
Place: Room 614, Education and Sciences Building, University of Haifa
Date and Time: January 7, 2025, 14:00-15:00
Recording: https://us02web.zoom.us/rec/share/galPb3quLg8KWpvmYl7f17Z_a0MdpWoTt0GgSxZ8iZWnQnnkR_robodACDzK5Kgh.nLVwgjqPXEdrB2xo
Title: Dense Ideals
Abstract: Around 1930 Banach and Ulam investigated the possibility of obtaining a sigma-additive probability measure extending the Lebesgue measure to all subsets of the real line. Banach and Kuratowski proved that the existence of such a measure is incompatible with the continuum hypothesis. Ulam extended this result by showing that indeed the existence of any nontrivial sigma-complete measure, measuring all subsets of some set, implies that either the continuum is large (the non-atomic case) or that there is a two-valued sigma-additive measure on an inaccessible cardinal (the atomic case).
Those two possibilities were eventually shown to be deeply connected by Solovay, in the sixties, using the methods of forcing and inner models. Extensions of the non-atomic case are called strong ideals, or generic large cardinals, and extensions of the atomic case are large cardinal axioms. The exact correspondence between those two is still vastly unknown. In this talk I will present a recent development in the area of strong ideals (joint with Monroe Eskew) and talk about some consequences.
