Dynamical infomorphism: Form of endo-perspective

The essential feature of the endo-perspective is examined, and a formal model of the endo-perspective is proposed by introducing the mixture of intra- and inter-operations. Because such a mixture in its naive realization entails a paradox within a formal system, we weaken the inter-operation in order to allow the formal system to be endowed with that mixture without a contradiction. The weakened inter-operation is related to the infomorphism proposed by Barwise [Information Flow, The Logic of Distributed Systems, Cambridge Univ. Press, 1997]. The formal model of the endo-perspective is thereby expressed as the dynamical infomorphism driven by that mixture. The endo-perspective is described as a formal system that includes the outside of the occupied perspective. If such an inclusion is applied to the common definition of a set, it entails Russel's paradox. Retaining the outside can be expressed as the mixture of the intent and the extent of a set together with the mixture of intra-operations within the intent (or the extent) and inter-operations between the intent and the extent. The endo-perspective, therefore, consists of two subsystems corresponding to the intent and the extent, respectively, and is defined as a system involving a particular mathematical tool (i.e., infomorphism) that allows for retaining the outside without a contradiction. Within that framework, the mixture of the intra- and the inter-operation drives the dynamical transition of the system, however, it can be terminated by its collapse. This collapse can be predicted from the internal logic defined within the system. The model is constructed through the verification of "a weakened paradox". Because the definition of the system involves a weakened paradox only, it does not always lead to a contradiction, although the collapse of the system corresponds to a contradiction. The double standards can be embedded into the system, the domain with truth-values (the inside) and the domain in which the collapse of the logic can occur (the outside).