Vol. 46, No. 2, 1973

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Peano models with many generic classes

James Henry Schmerl

Vol. 46 (1973), No. 2, 523–536
Abstract

The famous theorem of MacDowell and Specker asserts that every model of Peano arithmetic has a proper elementary end-extension. A consequence of their theorem (and its proof) is that every model of Peano arithmetic of cardinality less than κ has a κ-like elementary end-extension, and, in addition, if κ is regular, then there is such a κ-like model in which all classes are definable. However, under the assumption of the existence of a κ-Kurepa tree, each model of Peano arithmetic of cardinality less than κ does have a κ-like elementary end-extension in which there are more than κ generic classes.

Mathematical Subject Classification
Primary: 02H20
Milestones
Received: 22 December 1971
Revised: 10 July 1972
Published: 1 June 1973
Authors
James Henry Schmerl