In our talk, we will introduce two different notions of a spectrum of distributivity of forcings.

The first one is the fresh function spectrum, which is the set of regular cardinals \(\lambda\), such that the forcing adds a new function with domain \(\lambda\) all whose initial segments are in the ground model. We will provide several examples as well as general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing.

The second notion is the combinatorial distributivity spectrum, which is the set of possible regular heights of refining systems of maximal antichains without common refinement. We discuss the relation between the fresh function spectrum and the combinatorial distributivity spectrum. We consider the special case of \(\mathcal{P}(\omega)/fin\) (for which \(h\) is the minimum of the spectrum), and use a forcing construction to show that it is consistent that the combinatorial distributivity spectrum of \(\mathcal{P}(\omega)/fin\) contains more than one element.

This is joint work with Vera Fischer.