The Potential of Self-Reference

A second-order language - more precisely, any language rich enough to encode the Peano arithmetic - has the potential for self-reference. We have known since Gödel that this leads to the eventuality of true statements in the language that cannot be derived. But there has been little consideration of the practical upshot. If we view such languages as production systems: the expressivity of the language allows us to specify a set of rules for building and revising sets of production rules which constitute theories of the world. Self-reference allows for the possibility that this very set of production rules would be self-applicable, hence, self-revising. Thus, a finite set of axioms could be authored with an interpretation that nonetheless allowed for infinite adaptability, so to speak, over the long term. In second-order languages, the control afforded by rule sets need not be a fundamentally limitative feature.

The human intuition is that this infinite potential is in some way characteristic of living systems. Certainly ecologies evolving stochastically under conditions of natural selection evince it. And complex agents that learn and adapt within such ecologies may evince it, too.