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Claudio Conforti

lunes, 30 de julio de 2012

Forcing in Strategic Belief Models Fernando Tohme Gianluca Caterina e Rocco Gangle

Abstract
Forcing is a methodology for building models of Set Theory satisfying
certain properties. Since its inception by Paul Cohen, in the early 1960s,
it has been applied to several areas in Mathematical Logic, becoming a
powerful tool in the analysis of axiomatic systems. In this paper we extend
the applicability of forcing to game-theoretic strategic belief models. In
particular, we propose a very general notion of solutions for such games by
enlarging Brandenburger’s RmAR condition via extension through generic
types.


1
Introduction
The methodology of forcing was introduced into Mathematics by Paul Cohen in
order to show that Georg Cantor’s famous Continuum Hypothesis is independent
of the axioms of Zermelo-Frenkel Set Theory. This success prompted
other set theorists to investigate other topics in the field with the aid of this
powerful tool. Connections with other parts of Mathematical Logics were readily
found and versions of forcing for Model Theory were developed at the end of
the 1960s.
Forcing has remained in the realm of the foundations of Mathematics, with-
out being adopted in applied fields. The reason can be found in Shoenfield’s
Theorem, from which it can be deduced that forcing yields results only in the
non-absolute fragment of Mathematics, while most of applied science seems to
be confined in the absolute realm. Only two recent pieces of research dared
to go beyond this limit, in Design Theory and Abduction Theory.
In the latter, forcing is seen as providing the formal basis for diagrammatic
reasoning, embodied in Peirce’s γ-graphs. The intuition behind them seems to
extend to any belief formation process without defined boundaries.
A field in which the ideas of might be applied is the characterization
of types of players in games. While the conditions for the existence of com-
plete types spaces are fairly well known, we are interested in providing definite
features to the types that ensure the epistemic conditions for very general no-
tions of solution in games. This can be accomplished, we claim, by means of a
straightforward application of forcing.
In section 2 we present a conceptual discussion of Cohen’s variety of forcing
and how it allows to reason, from the point of view of a conceptual framework,
about generic objects in it and to provide a characterization of them, even if
they are indiscernable from inside the framework. In section 3 we make these
ideas concrete by introducing the problem of defining generic types in games
and apply forcing to define them.


1 comentario:

  1. Nos preparamos leyendo Forcing in Strategic Belief Models para el Homenaje de Horacio Arló Costa en Sadaf.
    Para quien lo desee tengo el artículo completo: conforti.claudio@gmail.com

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