Title: The Modal Logic of Forcing (Part III)
Abstract: Modal logic is used to study various modalities, i.e. various ways in which statements can be true, the most notable of which are the modalities of necessity and possibility. In set-theory, a natural interpretation is to consider a statement as necessary if it holds in any forcing extension of the world, and possible if it holds in some forcing extension. One can now ask what are the modal principles which captures this interpretation, or in other words - what is the "Modal Logic of Forcing"?
We can also restrict ourselves only to a certain class of forcing notions, or to forcing over a specific universe, resulting in an abundance of questions to be resolved.
We will begin with a short introduction to modal logic, and then present the tools developed by Joel Hamkins and Benedikt Loewe to answer these questions. We will present their answer to the original question, and then move to focus on the class of sigma-centered forcings, which I investigated in my Master's thesis.