Catching the strength of a theorem about chains with nice properties in posets of finite width

Marta Fiori Carones, Università di Udine. Part of the proofs, constructions and computations seminar series.

I will discuss the strength, from the point of view of reverse mathematics, of the following statement: for each countable poset P of finite width there is an infinite chain C such that each point of P is comparable with either none of the points in C or infinitely many points in C. Despite the fact that the original proof of this statement, given by Rival and Sands in the article "On the Adjacency of Vertices to the Vertices of an Infinite Subgraph",  goes through in Pi^1_1-CA0, we proved that the strength of this statement is much weaker. I will present some results and directions for further investigation. This is  work in progress joint with Alberto Marcone, Paul Shafer and Giovanni Soldà.