1. Under some (suitably weak) definition of consistency, the system \alpha can verify its own consistency.
2. The system \alpha is, in fact, consistent. (Requirement 2 is necessary because Godel's Incompleteness Theorem does not preclude an inconsistent axiom system from formally verifying a theorem asserting its own consistency.)
This talk will introduce two new axiom frameworks, called ISIT and ISITR. The first framework maps an arbitrary initial axiom system A onto an axiom system ISIT(A) which is self-verifying and can prove all the \Pi_1 theorems that A can prove. If conventional ZF-Set Theory is consistent, then our first Consistency-Preservation Theorem will establish that if A is consistent then ISIT(A) is consistent. It is not known whether ISITR is also a consistency-preserving framework, although the fundamental difference between ISIT and ISITR is essentially that the latter treats the sentences 4-2=2, 6-3=3, 8-4=4 .... as axioms rather than as theorems.
The surprising facet is that we can prove that if ISITR is, indeed, consistency-preserving, as one might conjecture, then P $\neq$ NP.
Our talk will explain why the relationship between ISIT and ISITR is much more puzzling than one might first anticipate. It will also conjecture that ISITR is consistency preserving.