Journal of Symbolic Logic

A Small Reflection Principle for Bounded Arithmetic

Rineke Verbrugge and Albert Visser

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We investigate the theory $I\Delta_0 + \Omega_1$ and strengthen [Bu86. Theorem 8.6] to the following: if $\mathrm{NP} \neq \mathrm{co-NP}$. then $\Sigma$-completeness for witness comparison formulas is not provable in bounded arithmetic. i.e. $I\delta_0 + \Omega_1 + \nvdash \forall b \forall c (\exists a(\operatorname{Prf}(a.c) \wedge \forall = \leq a \neg \operatorname{Prf} (z.b))\\ \rightarrow \operatorname{Prov} (\ulcorner \exists a(\operatorname{Prf}(a. \bar{c}) \wedge \forall z \leq a \neg \operatorname{Prf}(z.\bar{b})) \urcorner)).$ Next we study a "small reflection principle" in bounded arithmetic. We prove that for all sentences $\varphi$ $I\Delta_0 + \Omega_1 \vdash \forall x \operatorname{Prov}(\ulcorner \forall y \leq \bar{x} (\operatorname{Prf} (y. \overline{\ulcorner \varphi \urcorner}) \rightarrow \varphi)\urcorner).$ The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlak in [Pu86]. Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable $\Sigma$-completeness for witness comparison formulas.

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J. Symbolic Logic, Volume 59, Issue 3 (1994), 785-812.

First available in Project Euclid: 6 July 2007

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Verbrugge, Rineke; Visser, Albert. A Small Reflection Principle for Bounded Arithmetic. J. Symbolic Logic 59 (1994), no. 3, 785--812.

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