Journal of Symbolic Logic

Whither Relevant Arithmetic?

Harvey Friedman and Robert K. Meyer

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Based on the relevant logic $\mathrm{R}$, the system $\mathrm{R}^{\tt\#}$ was proposed as a relevant Peano arithmetic. $\mathrm{R}^{\tt\#}$ has many nice properties: the most conspicuous theorems of classical Peano arithmetic $\mathrm{PA}$ are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that $\mathrm{R}^{\tt\#}$ is properly weaker than $\mathrm{PA}$, in the sense that there is a strictly positive theorem $\mathrm{QRF}$ of $\mathrm{PA}$ which is unprovable in $\mathrm{R}^{\tt\#}$. The reason is interesting: if $\mathrm{PA}$ is slightly weakened to a subtheory $\mathrm{P}^+$, it admits the complex ring $\mathbf{C}$ as a model; thus $\mathrm{QRF}$ is chosen to be a theorem of $\mathrm{PA}$ but false in $\mathbf{C}$. Inasmuch as all strictly positive theorems of $\mathrm{R}^{\tt\#}$ are already theorems of $\mathrm{P}^+$, this nonconservativity result shows that $\mathrm{QRF}$ is also a nontheorem of $\mathrm{R}^{\tt\#}$. As a consequence, Ackermann's rule $\gamma$ is inadmissible in $\mathrm{R}^{\tt\#}$. Accordingly, an extension of $\mathrm{R}^{\tt\#}$ which retains its good features is desired. The system $\mathrm{R}^{\tt\#}{\tt\#}$, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between $\mathrm{R}^{\tt\#}$ and $\mathrm{R}^{\tt\#}{\tt\#}$, which does formalize arithmetic on relevant principles, but also admits $\gamma$ in a natural way?

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J. Symbolic Logic, Volume 57, Issue 3 (1992), 824-831.

First available in Project Euclid: 6 July 2007

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Friedman, Harvey; Meyer, Robert K. Whither Relevant Arithmetic?. J. Symbolic Logic 57 (1992), no. 3, 824--831.

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