Provability and Decidability of Arithmetical Sentences

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).