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Provability and Decidability of Arithmetical Sentences

  • Shih Ping TungEmail author
Conference paper
  • 115 Downloads
Part of the Logic in Asia: Studia Logica Library book series (LIAA)

Abstract

Gödel’s incompleteness theorem tells that the elementary theories of \({\mathbf N}\) and \({\mathbf Z}\) are not axiomatizable, respectively. J. Robinson proved that the elementary theory of \({\mathbf Q}\) is undecidable, hence not axiomatizable. We may ask what subsets of these elementary theories are decidable or axiomatizable. We call an arithmetical sentence \(\phi \) an \(\exists ^n\forall \exists \) sentence if it is logically equivalent to a sentence of the form \(\exists x_1\cdots \exists x_n\forall y\exists z \,\psi (x_1, \ldots ,x_n,y,z),\) where \(\psi (x_1, x_2,\ldots ,x_n,y,z)\) is a quantifier-free formula. In this talk we show that the set of all true \(\exists ^n\forall \exists \) sentences in \(\mathbf N,\) \(\mathbf Z\), and \(\mathbf Q\) are axiomatizable, respectively. Also, let \(f(x,y)\in {\mathbf Z}[x,y]\) and \(a\in {\mathbf Z}\). The sets of all sentences of the form \(\forall z\exists x \forall y\, f(x,y)-az \ne 0\) true in \(\mathbf N,\) \(\mathbf Z\), and \(\mathbf Q\) are also axiomatizable, respectively. Therefore, the sets of sentences of the form \(\exists z\forall x \exists y\, f(x,y)-az =0\) true in \(\mathbf N,\) \(\mathbf Z\), and \(\mathbf Q\) are decidable, respectively. Over \(\mathbf N\), we have shown that this decision problem is NP-hard, co-NP-hard, and in PSPACE. Therefore, if NP \(\ne \) co-NP, then this decision problem is in PSPACE \(\setminus \) (NP \(\cup \) co-NP).

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Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsChung Yuan Christian UniversityTaoyuanTaiwan

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