# Sh:767

- Shelah, S., & Tsuboi, A. (2002).
*Definability of initial segments*. Notre Dame J. Formal Logic,**43**(2), 65–73 (2003). arXiv: math/0104277 DOI: 10.1305/ndjfl/1071509428 MR: 2033316 -
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. - 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}, doi = {10.1305/ndjfl/1071509428}, mrclass = {03C62 (03C55 03H15)}, mrnumber = {2033316}, mrreviewer = {Roman Kossak}, doi = {10.1305/ndjfl/1071509428}, note = {\href{https://arxiv.org/abs/math/0104277}{arXiv: math/0104277}}, arxiv_number = {math/0104277} }