Resource Limited Theories and their Extensions

Abstract

This work is based on the idea that extension of physical and mathematical theories to include the amount of space, time, momentum, and energy resources required to determine properties of systems may influence what is true in physics and mathematics at a foundational level. Background material, on the dependence of region or system sizes on both the resources required to study the regions or systems and the indirectness of the reality status of the systems, suggests that one associate to each amount, r, of resources a domain, D_{r}, a theory, T_{r}, and a language, L_{r}. D_{r} is limited in that all statements in D_{r} require at most r resources to verify or refute. T_{r} is limited in that any theorem of T_{r} must be provable using at most r resources. Also any theorem of T_{r} must be true in D_{r}. L_{r} is limited in that all expressions in L_{r} require at most r resources to create, display, and maintain. A partial ordering of the resources is used to describe minimal use of resources, a partial ordering of the T_{r}, and motion of an observer using resources to acquire knowledge. Reflection principles are used to push the effect of Godel's incompleteness theorem on consistency up in the partial ordering. It is suggested that a coherent theory of physics and mathematics, or theory of everything, is a common extension of all the T_{r}.