The exile of (certain kinds of open caveat or relatively innocuous) contradictory statements in foundational theory formulation merely delays and displaces the onset of various discontinuities and anomalies for a later date, evident where any theory may begin to fail to adequately or comprehensively explain the observed complexities of reality; for instance, where sufficiently sophisticated descriptive systems encounter their own explanatory frameworks as elements of the systems they seek to explain.
Mature theoretical foundations may not in every instance be those seeking a pure binary truth-value completeness so much as an implicit orientation (or at the very least a pre-emptive view) to probable future theoretical extensibility and progressive theory reformulation. Any science of dynamic complex systems might find unexpected utility in recursive application to it’s own theoretical foundations and through this – a self-conscious insight into the process of complex systems theory development itself.
If only there were an autonomous, logical way to assess axioms and lemmas at their introduction into explanatory systems in regards to their probability of eventual failure, saving a large amount of wasted time, investment and eventual rollback…
Daedelus Kite 