Results about sets of desirable gamble sets

Coherent sets of desirable gamble sets is used as a model for representing an agents opinions and choice preferences under uncertainty. In this paper we provide some results about the axioms required for coherence and the natural extension of a given set of desirable gamble sets. We also show that coherent sets of desirable gamble sets can be represented by a proper filter of coherent sets of desirable gambles. This paper was primarily written in 2021, overlapping with my finishing up Campbell-Moore (2021). There is some overlap between this paper and de Cooman et al. (2023); the results of this paper were shown independently.