This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s unramified (...) forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

At the beginning of this article I propose to use the word “ evolution “ as it is used in biology, to mean the formation of a number of vegetable or animal species out of a few comparatively simple types, and to exclude from its connotation any idea of perfection, purpose, value, and so on.

Author's answers to questions on Buddhist philosophy, posed to him by some foreign scholars.

It is universally held and in all classical Greek Grammars taught that in their numeral system, the ancient Greeks denoted the figure for 6 by the digamma or vau, that for 90 by the koppa, and that for 900 by the sampi or sanpi. On a closer examination, however, this doctrine proves erroneous and requires correction.

On the philosophy of Sir Muhammad Iqbal, 1877-1938 with special reference to religious experiences and philosophy.