Posted by Anton Hughes on Sunday, August 26, 2007 with No comments
"One could make the same opening statement on the "obviousness" of TNT's consistency as Imprudence did in regard to the Propositional Calculus, namely, that each rule embodies a reasoning principle which we fully believe in, and therefore to question the consistency of TNT is to question our own sanity. To some extent this argument still carries weight - but not quite so much weight as before. There are just too many rules of inference, and some of them just might be slightly 'off'. Furthermore, how do we know that this mental model we have of some abstract entities called "natural numbers" is actually a coherent construct? Perhaps our own thought processes, which we have tried to capture in the formal rules of the system, are themselves inconsistent!!It is of course not the kind of thing we expect, but it gets more and more conceivable that our thoughts might lead us astray, the more complex the subject matter gets..."