Logic & Set Theory seminar: The tree property and the continuum function
Radek Honzik, Charles University, Prague
Howard House 4th floor seminar room
We will review some of the recent results concerning the tree property in connection to the values of 2κ for κ in the neighbourhood of the cardinal with the tree property. In particular, we will construct a model where 2κ is large, κ is a singular strong limit cardinal of cofinality ω, and the tree property holds at κ++. We will discuss potential problems in adding collapses to make κ = ℵω. Finally, we show that the failure of weak square (i.e. the non-existence of special Aronszajn trees) at every ℵn, 1 < n < ω, is consistent with an arbitrary continuum function below ℵω which violates GCH at every ℵn, 0 ≤ n < ω. The work is joint with Sy Friedman and Sarka Stejskalova.