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

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
Received: 22 December 1971
Revised: 10 July 1972
Published: 1 June 1973
James Henry Schmerl