Local sentences and Mahlo cardinals
From MaRDI portal
Publication:5434200
DOI10.1002/MALQ.200610049zbMath1130.03023arXiv0705.2298OpenAlexW2024325951MaRDI QIDQ5434200
Stevo Todorčević, Olivier Finkel
Publication date: 4 January 2008
Published in: Mathematical Logic Quarterly (Search for Journal in Brave)
Abstract: Local sentences were introduced by J.-P. Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P. Ressayre, Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ]. Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In fact, we show that the stretching principles are equivalent to the existence of n-Mahlo cardinals for appropriate integers n. This is done by proving first that for each integer n, there is a local sentence phi_n which has well ordered models of order type alpha, for every infinite ordinal alpha > omega which is not an n-Mahlo cardinal.
Full work available at URL: https://arxiv.org/abs/0705.2298
Consistency and independence results (03E35) Large cardinals (03E55) Model theory of finite structures (03C13) Model theory of ordered structures; o-minimality (03C64) Set-theoretic model theory (03C55)
Cites Work
Related Items (1)
This page was built for publication: Local sentences and Mahlo cardinals
