# Sh:767

• Shelah, S., & Tsuboi, A. (2002). Definability of initial segments. Notre Dame J. Formal Logic, 43(2), 65–73 (2003).
• 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 §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 §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.
• Version 2003-05-19_11 (11p) published version (9p)
Bib entry
@article{Sh:767,
author = {Shelah, Saharon and Tsuboi, Akito},
title = {{Definability of initial segments}},
journal = {Notre Dame J. Formal Logic},
fjournal = {Notre Dame Journal of Formal Logic},
volume = {43},
number = {2},
year = {2002},
pages = {65--73 (2003)},
issn = {0029-4527},
mrnumber = {2033316},
mrclass = {03C62 (03C55 03H15)},
doi = {10.1305/ndjfl/1071509428},
note = {\href{https://arxiv.org/abs/math/0104277}{arXiv: math/0104277}},
arxiv_number = {math/0104277}
}