How to Extend The Semantic Tableaux and Cut-Free Versions of The Second Incompleteness Theorem Almost to Robinson's Arithmetic Q

Let us recall that Raphael Robinson’s Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13, 18]. We will generalize the semantictableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers and denoted as V , such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)