On Wilkie and Paris’s notion of fullness
Volume 245 / 2019
Fundamenta Mathematicae 245 (2019), 79-100
MSC: Primary 03C62; Secondary 03H15.
DOI: 10.4064/fm491-7-2018
Published online: 14 December 2018
Abstract
In their comprehensive study of the End-Extension Question, Wilkie and Paris devised a special notion of saturation called \emph {fullness} for models of arithmetic (Logic, Methodology and Philosophy of Science VIII, North-Holland, 1989, pp. 143–161). In this paper, we take a closer look at that notion of fullness and refine some results in the Wilkie–Paris paper. In particular, we characterize fullness in terms of the existence of recursively saturated end extensions. From this we deduce that every countable $\mathrm {I}\Delta _0$-full model of $\mathrm {I}\Delta _0+\mathrm {B}\Sigma _1$ is $(\mathrm {I}\Delta _0+\mathrm {B}\Sigma _1)$-full.
