Separation of bounded arithmetic using a consistency statement

Abstract

This paper proves Buss's hierarchy of bounded arithmetics $S^1_2 \subseteq S^2_2 \subseteq \cdots \subseteq S^i_2 \subseteq \cdots$ does not entirely collapse. More precisely, we prove that, for a certain $D$, $S^1_2 \subsetneq S^{2D+5}_2$ holds. Further, we can allow any finite set of true quantifier free formulas for the BASIC axioms of $S^1_2, S^2_2, \ldots$. By Takeuti's argument, this implies $\mathrm{P} \neq \mathrm{NP}$. Let $\mathbf{Ax}$ be a certain formulation of BASIC axioms. We prove that $S^1_2 \not\vdash \mathrm{Con}(\mathrm{PV}^-_1(D) + \mathbf{Ax})$ for sufficiently large $D$, while $S^{2D+7}_2 \vdash \mathrm{Con}(\mathrm{PV}^-_1(D) + \mathbf{Ax})$ for a system $\mathrm{PV}^-_1(D)$, a fragment of the system $\mathrm{PV}^-_1$, induction free first order extension of Cook's $\mathrm{PV}$, of which proofs contain only formulas with less than $D$ connectives. $S^1_2 \not\vdash \mathrm{Con}(\mathrm{PV}^-_1(D) + \mathbf{Ax})$ is proved by straightforward adaption of the proof of $\mathrm{PV} \not\vdash \mathrm{Con}(\mathrm{PV}^-)$ by Buss and Ignjatovi\'c. $S^{2D+5}_2 \vdash \mathrm{Con}(\mathrm{PV}^-_1(D) + \mathbf{Ax})$ is proved by $S^{2D+7}_2 \vdash \mathrm{Con}(\mathrm{PV}^-_q(D+2) + \mathbf{Ax})$, where $\mathrm{PV}^-_q$ is a quantifier-only extension of $\mathrm{PV}^-$. The later statement is proved by an extension of a technique used for Yamagata's proof of $S^2_2 \vdash \mathrm{Con}(\mathrm{PV}^-)$, in which a kind of satisfaction relation $\mathrm{Sat}$ is defined. By extending $\mathrm{Sat}$ to formulas with less than $D$-quantifiers, $S^{2D+3}_2 \vdash \mathrm{Con}(\mathrm{PV}^-_q(D) + \mathbf{Ax})$ is obtained in a straightforward way.