zbMATH — the first resource for mathematics

A gap 1 cardinal transfer theorem. (English) Zbl 1095.03051
Summary: We extend the gap 1 cardinal transfer theorem $$(\kappa^+,\kappa) \to (\lambda^+,\lambda)$$ to any language of cardinality $$\leq \lambda$$, where $$\lambda$$ is a regular cardinal. This transfer theorem has been proved by Chang under GCH for countable languages and by Silver in some cases for bigger languages (also under GCH). We assume the existence of a coarse $$(\lambda, 1)$$-morass instead of GCH.

MSC:
 03E45 Inner models, including constructibility, ordinal definability, and core models 03C55 Set-theoretic model theory 03E05 Other combinatorial set theory 03C80 Logic with extra quantifiers and operators 03C50 Models with special properties (saturated, rigid, etc.) 03E35 Consistency and independence results 03E65 Other set-theoretic hypotheses and axioms
Full Text:
References:
 [1] Chang, Proc. Amer. Math. Soc. 16 pp 1148– (1965) [2] and , Model Theory, 3rd ed. (North-Holland, 1993). [3] Constructibility (Springer-Verlag, 1984). · Zbl 0542.03029 [4] Coarse morasses in L . In: Lecture Notes in Mathematics 872, pp. 37–54 (Springer-Verlag, 1981). [5] and , Primitive recursive set functions. In: Axiomatic Set Theory (D. Scott, ed.). Proceedings of Symposia in Pure Mathematics 13, pp. 143–167 (AMS, 1971). [6] Jensen, Ann. Math. Logic 4 pp 229– (1972) [7] private communication. [8] The ({$$\kappa$$}, {$$\beta$$} )-morass. Unpublished manuscript. [9] Kennedy, J. Symbolic Logic 67 pp 1169– (2002) [10] , and , Weakly compact cardinals and {$$\kappa$$} -torsionless modules. In preparation. · Zbl 1214.03034 [11] Higher Recursion Theory (Springer-Verlag, 1990). · Zbl 0716.03043 [12] A short course on gap-one morasses with a review of the fine structure of L . In: Surveys in Set Theory, pp. 197–243. London Math. Soc. Lecture Note Series 87 (Cambridge University Press, 1983). [13] The two cardinal transfer theorem for languages of arbitrarily cardinality. Submitted to the Journal of Symbolic Logic. [14] Über zwei-Kardinalzahl-Probleme. Diplomarbeit, Universität Freiburg, 1981.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.