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

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Goedel's Incompleteness Theorem is essentially a 2-part result, whosesecond facet establishes that all conventional axiom systems are unable toprove a theorem corroborating their own consistency. This latter effect,called the Second Incompleteness Theorem, is quite counter-intuitivebecause human beings seem to implicitly believe that their cogitationprocesses are consistent (in order for them to gain the needed motivationalenergy for cogitation). Yet quite surprisingly, the Second IncompletenessTheorem has demonstrated that conventional logics cannot corroborate such aseemingly very natural consistency assumption.The main goal of Willard's research project will be to explore to whatextent a logic formalism can wiggle around the Second IncompletenessTheorem and formalize a type of at least partial instinctive faith inits own consistency. Willard has already published several articlesexploring boundary-case exceptions to the Second IncompletenessTheorem and generalizations of it, thus establishing the basicfoundational concepts for this quite unorthodox research approachWillard's planned investigation will have an inter-disciplinaryemphasis, being germane to computer science, mathematics, thecognitive and information sciences, philosophy and linguistics. Itwill show that while self-justifying axiom systems have a veryunconventional outer shell, they do retain an ability to simulate mostof the practical computer-oriented knowledge of the data needed by anyarbitrary logically valid axiom system. It will also show howcomputerized floating-point real number arithmetics behave verydifferently from integer arithmetics and how self-justifyingformalisms retain an ability to expand their knowledge base by hoppingand skipping form one formal self-justifying axiom system to another(via the use of Mind Change Theory and sundry trial-and-errorexperimental methodologies).The Second Incompleteness Theorem is sufficiently robust that itsevasion is feasible only in a limited context, where the specifiedlogic possesses a component that is unconventional in a nontrivialway. The reason that such unconventional logics are of interest isthat human beings evidently possess some type of innate instinctiveunderstanding of their own self-consistencies (at least under someformal definitions of consistency). It is therefore desirable tounderstand what type of algorithmic processes and foundationalprecepts from symbolic logic can simulate a Thinking Being'sconception of its own consistency (in either a full or partial sense).The goal of this research project will be to investigate the nature ofand the potential uses and implications of logics that possess sometype of well-defined partial knowledge of their own consistency.