EAGER: An Investigation of the Partial Degrees in Which Logics Can Recognize Their Own Consistency and the Potentially Broad Inter-Disciplinary Implications of These Effects

Suny At Albany, Albany NY

Investigators

Abstract

Goedel's Incompleteness Theorem is essentially a 2-part result, whose second facet establishes that all conventional axiom systems are unable to prove a theorem corroborating their own consistency. This latter effect, called the Second Incompleteness Theorem, is quite counter-intuitive because human beings seem to implicitly believe that their cogitation processes are consistent (in order for them to gain the needed motivational energy for cogitation). Yet quite surprisingly, the Second Incompleteness Theorem has demonstrated that conventional logics cannot corroborate such a seemingly very natural consistency assumption. The main goal of Willard's research project will be to explore to what extent a logic formalism can wiggle around the Second Incompleteness Theorem and formalize a type of at least partial instinctive faith in its own consistency. Willard has already published several articles exploring boundary-case exceptions to the Second Incompleteness Theorem and generalizations of it, thus establishing the basic foundational concepts for this quite unorthodox research approach Willard's planned investigation will have an inter-disciplinary emphasis, being germane to computer science, mathematics, the cognitive and information sciences, philosophy and linguistics. It will show that while self-justifying axiom systems have a very unconventional outer shell, they do retain an ability to simulate most of the practical computer-oriented knowledge of the data needed by any arbitrary logically valid axiom system. It will also show how computerized floating-point real number arithmetics behave very differently from integer arithmetics and how self-justifying formalisms retain an ability to expand their knowledge base by hopping and skipping form one formal self-justifying axiom system to another (via the use of Mind Change Theory and sundry trial-and-error experimental methodologies). The Second Incompleteness Theorem is sufficiently robust that its evasion is feasible only in a limited context, where the specified logic possesses a component that is unconventional in a nontrivial way. The reason that such unconventional logics are of interest is that human beings evidently possess some type of innate instinctive understanding of their own self-consistencies (at least under some formal definitions of consistency). It is therefore desirable to understand what type of algorithmic processes and foundational precepts from symbolic logic can simulate a Thinking Being's conception of its own consistency (in either a full or partial sense). The goal of this research project will be to investigate the nature of and the potential uses and implications of logics that possess some type of well-defined partial knowledge of their own consistency.