Definability of initial segments
Abstract
We consider implicit definability of the standard part {0,1,...} in nonstandard models of Peano arithmetic (PA), and we ask whether there is a model of PA in which the standard part is implicitly definable. In section 1, we define a certain class of formulas, and show that in any model of PA the standard part is not implicitly defined by using such formulas. In section 2 we construct a model of PA in which the standard part is implicitly defined. To construct such a model, first we assume a set theoretic hypothesis diamondsuit_{S_lambda^{lambda^+}}, which is an assertion of the existence of a very general set. Then we shall eliminate the hypothesis using absoluteness for the existence of a model having a tree structure with a certain property.
Keywords
Cite
@article{arxiv.math/0104277,
title = {Definability of initial segments},
author = {Saharon Shelah and Akito Tsuboi},
journal= {arXiv preprint arXiv:math/0104277},
year = {2007}
}