How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q

Journal of Symbolic Logic 67 (1):465-496 (2002)
Abstract
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 semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 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 IΣ 0 , as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)
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DOI 10.2178/jsl/1190150055
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References found in this work BETA
On the Scheme of Induction for Bounded Arithmetic Formulas.A. J. Wilkie & J. B. Paris - 1987 - Annals of Pure and Applied Logic 35 (3):261-302.
On Herbrand Consistency in Weak Arithmetic.Zofia Adamowicz & Paweł Zbierski - 2001 - Archive for Mathematical Logic 40 (6):399-413.

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Passive Induction and a Solution to a Paris–Wilkie Open Question.Dan E. Willard - 2007 - Annals of Pure and Applied Logic 146 (2):124-149.

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