An introduction to forcing axioms, SRP and OCA

TL;DR

This paper introduces forcing axioms, focusing on SRP and OCA, discussing their definitions, consequences, and applications in set theory, especially related to the continuum and combinatorial structures.

Contribution

It provides a concise overview of forcing axioms, their equivalences, and explores the consequences of SRP and OCA, including consistency proofs and applications.

Findings

SRP follows from MM and has significant reflection properties.

OCA acts as a two-dimensional perfect set property for analytic sets.

OCA has applications in continuum properties, such as gaps in ω^ω.

Abstract

These notes are extracted from the lectures on forcing axioms and applications held by professor Matteo Viale at the University of Turin in the academic year 2011-2012. Our purpose is to give a brief account on forcing axioms with a special focus on some consequences of them (SRP, OCA, PID). These principles were first isolated by Todor\v cevi\'c and interpolate most consequences of MM and PFA, thus providing a useful insight on the combinatorial structure of the theory of forcing axioms. In the first part of this notes we will give a brief account on forcing axioms, introducing some equivalent definition by means of generalized stationarity and presenting the consequences of them in terms of generic absoluteness. In the second part we will state the strong reflection principle (SRP), prove it under MM and examine its main consequences. This axiom is defined in terms of reflection properties of generalized stationary set. In the third part we will state the open coloring axiom (OCA), and provide consistency proofs for some versions of it. This axiom can be seen as a sort of two-dimensional perfect set property, i.e. the basic descriptive set theory result that every analytic set is either countable or it contains a perfect subset. In the last part we will explore a notable application of OCA to problems concerning properties of the continuum, in particular the existence of certain kind of gaps in $\omega^\omega$.