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)
Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms , . 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 .)
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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.
Citations of this work BETA
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.
A Generalization of the Second Incompleteness Theorem and Some Exceptions to It.Dan E. Willard - 2006 - Annals of Pure and Applied Logic 141 (3):472-496.
Unifying the Model Theory of First-Order and Second-Order Arithmetic viaWKL0⁎.Ali Enayat & Tin Lok Wong - 2017 - Annals of Pure and Applied Logic 168 (6):1247-1283.
Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories.Saeed Salehi - 2013 - Archive for Mathematical Logic 52 (3-4):317-333.
2005 Summer Meeting of the Association for Symbolic Logic. Logic Colloquium'05.Stan S. Wainer - 2006 - Bulletin of Symbolic Logic 12 (2):310-361.
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