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Local sentences and Mahlo cardinals
Author(s) -
Finkel Olivier,
Todorcevic Stevo
Publication year - 2007
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200610049
Subject(s) - mathematics , equivalence (formal languages) , axiom , regular cardinal , sentence , consistency (knowledge bases) , combinatorics , integer (computer science) , order (exchange) , discrete mathematics , type (biology) , pure mathematics , computer science , artificial intelligence , geometry , ecology , finance , economics , biology , programming language
Local sentences were introduced by Ressayre in [6] 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 [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. 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 every integer n , there is a local sentence φ n having well ordered models of order type τ , for every infinite ordinal τ > ω which is not an n ‐Mahlo cardinal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)