Hypocomputation

TL;DR

This paper explores the structural differences between traditional computational theories and hypercomputational theories, providing a survey of known landmarks to aid in developing hypercomputational frameworks.

Contribution

It offers a survey of the structural landscape of formal theories, highlighting the transition from computational to hypercomputational theories and addressing gaps in current metamathematical research.

Findings

Identifies key landmarks in formal theories

Highlights the structural differences between computational and hypercomputational theories

Provides guidance for developing hypercomputational theories

Abstract

Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of metamathematics, and so many of the previous results have been folded into other theories, that we are in danger of loosing an appreciation of the broader structure of formal theories. As an aid to those looking to develop hypercomputational theories, we will briefly survey the known landmarks both inside and outside the borders of computational theory. We will not focus in this paper on why the structure of formal theory looks the way it does. Instead we will focus on what this structure looks like, moving from hypocomputational, through traditional computational theories, and then beyond to hypercomputational theories.