Existence and definability of states of the world

Abstract

We present here a notion of state of the world general enough to embrace most circular phenomena in economics and game theory. We prove that it obtains by unfolding beliefs only if the process of belief generation has a fixed point. Otherwise, we are led to an unending transfinite hierarchy. This result indicates that Zermelo-Frenkel's set theory cannot provide the modeling tools for the representation of states of the world. We apply, instead, a theory of non-well-founded sets. In that framework, states of the world are legitimate objects which can be seen as fixed points of the belief-generation operator. Not every possible state of the world can be unfolded in a hierarchy of beliefs. It will be shown by means of a simple argument, based in Tarski's indefinability theorem, that there exist states of the world that are not expressible in that way. Moreover, this result implies that there is no way to represent those states of the world in a consistent language. However, if we assume agents do not have negative self-referential beliefs, the unfolding of beliefs suffices.