Mathematics selfproves its own Consistency (contra Gödel et. al.)
Carl Hewitt  International Society for Inconsistency Robustness  [Home Page] 
Notice: Hosted by Richard Waldinger
Date: Tuesday, April 24th 2012 at 4:00pm
Location: EJ228 (SRI E building) (Directions)

The consistency of mathematics has long been of concern. Using selfreferential propositions, Gödel et. al. proved that if mathematics is consistent, it does not prove its own consistency. Regardless, this paper presents a simple nonconstructive selfproof of consistency:
Of course, the above proof does not show that mathematics is really consistent, i.e., that is impossible to infer an inconsistency using the inference rules because the proof is valid even if mathematics is inconsistent. In fact, selfproving consistency raises that possibility that mathematics could be inconsistent because of contradiction with the result of Gödel et. al. One resolution is not to have selfreferential propositions. This can be achieved by carefully arranging the rules for propositions so that selfreferential propositions cannot be constructed. Fortunately, selfreferential propositions do not seem to have any important practical applications. The above proof of consistency is carried out in Direct Logic [Hewitt 2011] that is a powerful inconsistencyrobust inference system that is its own metasystem. Having such a system is important in computer science because computers must be able to carry out all inferences (including inferences about their own inference processes) without always relying on humans. A tradeoff is that in return for having such a powerful inference system, selfreferential propositions are not allowed. 

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