2001/04/28 by Saharon Shelah, Shelah, Saharon, Akito Tsuboi +1
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #History and Theory of Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO
paper · pdf · doi:10.48550/arxiv.math/0104277
published as Notre Dame J. Formal Logic 43 No. 2 (2002) 65--73 (2003)
arxiv created 2001/04/28 · openalex publication_date 2001/04/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 diamondsuitSlambdalambda+, 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.