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| '''Self-verifying theories''' are consistent [[first-order logic|first-order]] systems of [[arithmetic]] much weaker than [[Peano arithmetic]] that are capable of proving their own [[consistency proof|consistency]]. [[Dan Willard]] was the first to investigate their properties, and he has described a family of such systems. According to [[Gödel's incompleteness theorem]], these systems cannot contain the theory of Peano arithmetic, and in fact, not even the weak fragment of [[Robinson arithmetic]]; nonetheless, they can contain strong theorems; for instance there are self-verifying systems capable of proving the consistency of Peano arithmetic.
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| In outline, the key to Willard's construction of his system is to formalise enough of the [[Gödel]] machinery to talk about [[provability]] internally without being able to formalise [[Diagonal lemma|diagonalisation]]. Diagonalisation depends upon being able to prove that multiplication is a total function (and in the earlier versions of the result, addition also). Addition and multiplication are not function symbols of Willard's language; instead, subtraction and division are, with the addition and multiplication predicates being defined in terms of these. Here, one cannot prove the [[arithmetical hierarchy|<math>\Pi^0_2</math> sentence]] expressing totality of multiplication:
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| :<math>(\forall x,y)\ (\exists z)\ {\rm multiply}(x,y,z).</math> | |
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| where <math>{\rm multiply}</math> is the three-place predicate which stands for <math>z/y=x</math>.
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| When the operations are expressed in this way, provability of a given sentence can be encoded as an arithmetic sentence describing termination of an [[analytic tableau]]. Provability of consistency can then simply be added as an axiom. The resulting system can be proven consistent by means of a [[relative consistency]] argument with respect to ordinary arithmetic.
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| We can add any true <math>\Pi^0_1</math> sentence of arithmetic to the theory and still remain consistent.
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| {{logic-stub}}
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| ==References==
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| *Solovay, R., 1989. "Injecting Inconsistencies into Models of PA". Annals of Pure and Applied Logic 44(1-2): 101—132.
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| *Willard, D., 2001. "Self Verifying Axiom Systems, the Incompleteness Theorem and the Tangibility Reflection Principle". Journal of Symbolic Logic 66:536—596.
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| *Willard, D., 2002. "How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem to Robinson's Arithmetic Q" . Journal of Symbolic Logic 67:465—496.
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| ==External links==
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| * [http://www.cs.albany.edu/FacultyStaff/profiles/willard.htm Dan Willard's home page].
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| [[Category:Proof theory]]
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| [[Category:Theories of deduction]]
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