0100492302

open access

Accepted Manuscript

On collection schemes and Gaifman's splitting theorem

English (英語)

Mathematical Logic Quarterly

70(4)

398-413

John Wiley & Sons

2024-11

2025-11-10

Published Online: 2024-11-10

We study model theoretic characterizations of various collection schemes over PA− from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any n≥0 and M⊨PA−, the following are equivalent: 1. M satisfies the collection scheme for Σn+1 formulas. 2. For any K,N⊨PA−, if M⊆cofK, M≺Δ0K and M≺N, then M≺Σn+2K and supN(M)≺ΣnN. 3. For any N⊨PA−, if M≺N, then M≺Σn+2supN(M)≺ΣnN. Here, supN(M) is the unique K satisfying M⊆cofK⊆endN. We also investigate strong collection schemes and parameter-free collection schemes from the similar perspective.

学術雑誌論文

This is the peer reviewed version of the following article: [Kurahashi, T. and Minami, Y. (2024), On collection schemes and Gaifman's splitting theorem. Math. Log. Quart., 70: 398-413.], which has been published in final form at [https://doi.org/10.1002/malq.202400021]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.