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According to Hewitt, in concrete terms for Actor systems, typically we cannot observe the details by which the arrival order of messages for an Actor is determined. Attempting to do so affects the results and can even push the indeterminacy elsewhere. e.g., see [[metastability in electronics]] and [[arbiter (electronics)|arbiters]]. Instead of observing the internals of arbitration processes of Actor computations, we await outcomes. Indeterminacy in arbiters produces indeterminacy in Actors. The reason that we await outcomes is that we have no alternative because of indeterminacy.
It is important to be clear about the basis for the published claim about the limitation of mathematical logic. It was not just that Actors could not, in general, be implemented in mathematical logic. The published claim was that because of the indeterminacy of the physical basis of the Actor model, that no kind of deductive mathematical logic could escape the limitation. This became important later when researchers attempted to extend [[Prolog]] (which had some basis in [[logic programming]]) to concurrent computation using message passing. (See the section below).
What does the mathematical theory of Actors have to say about this? A ''closed'' system is defined to be one which does not communicate with the outside. [[Actor model theory]] provides the means to characterize all the possible computations of a closed Actor system using the Representation Theorem [Hewitt 2007] as follows:
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