Are all natural numbers the same

TL;DR

This paper surveys recent efforts and challenges in extending the proper forcing axiom (PFA) to higher infinite cardinals, highlighting key developments and obstacles in this area of set theory.

Contribution

It provides a comprehensive overview of the state-of-the-art and identifies difficulties in developing higher analogues of PFA for aleph_n cardinals.

Findings

Several attempts have been made to extend PFA to aleph_n for n > 1

Difficulties in constructing higher analogues of PFA are significant

Some successes in developing higher forcing axioms are surveyed

Abstract

This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form aleph_n where n > 1. We investigate the difficulties of doing this and survey some of the successes