Weakly Represented Families in the Context of Reverse Mathematics

It is common pratice in computability theory to represent a family of objects by a single object; this is typically done in such a way that the individual members of the family can be derived in a uniformly effective way from the single object. For example, a single set A could represent the family of sets consisting of the rows of A. This approach is called uniform and is relatively restricted in what families of objects it allows to represent. In this article we study the reverse mathematics proof strength of second order logic principles that make statements about families of sets and functions. To allow us to make these statements more expressive, we would like to formulate them in such a way that they talk about a larger class of families of sets and functions than can be represented in the uniform way. To enable this, we define a more general way of representing families, the so-called weak representation of families of functions and sets. Using this tool, we can then state and investigate the second order logic principles we want to study. Specifically, we investigate the Domination Principle DOM, the Avoidance Principle AVOID, the Meeting Principle MEET, and the Hyperimmunity Principle HI. Furthermore, we define the Cohesion Principle for weakly represented families COHW and separate it from the previously known Cohesion Principle COH. The results obtained witness that the notion of weakly represented families is a useful and robust tool in reverse mathematics. ? D. Raghavan was supported in part by NUS grants R146-000-184-112. ?? F. Stephan was supported in part by NUS grants R146-000-181-112 and R146-000184-112.