Minimal elementary extensions of models of set theory and arithmetic

We discuss minimal elementary extensions of models of set theory and contrast the behavior of models of set theory and arithmetic as regarding such extensions. Our main result, proved using a Boolean ultrapower argument, is:TheoremEvery model of ZFChas a conservative elementary extension which possesses a cofinal minimal elementary extension.An application of Boolean ultrapowers to models of full arithmetic is also presented.

[1]  Ali Enayat.  UNDEFINABLE CLASSES AND DEFINABLE ELEMENTS IN MODELS OF SET THEORY AND ARITHMETIC , 1988 .

[2]  Klaus Potthoff.  A simple tree lemma and its application to a counterexample of philips , 1977, Arch. Math. Log..

[3]  Andreas Blass.  Conservative extensions of models of arithmetic , 1980, Arch. Math. Log..

[4]  Haim Gaifman,et al.  Models and types of Peano's arithmetic , 1976 .

[5]  R. G. Phillips.  Omitting types in arithmetic and conservative extensions , 1974 .

[6]  R. Mansfield.  The theory of Boolean ultrapowers , 1971 .

[7]  John L. Bell,et al.  Boolean-valued models and independence proofs in set theory , 1977 .

[8]  George Mills.  A Model of Peano Arithmetic with no Elementary End Extension , 1978, J. Symb. Log..

[9]  Julia F. Knight.  Hanf Numbers for Omitting Types Over Particular Theories , 1976, J. Symb. Log..

[10]  Ali Enayat.  Conservative Extensions of Models of Set Theory and Generalizations , 1986, J. Symb. Log..

[11]  Kenneth Kunen.  Some points in N , 1976 .

[12]  A. Blass.  End extensions, conservative extensions, and the Rudin-Frolík ordering , 1977 .

[13]  E. Specker,et al.  Modelle der Arithmetik , 1990 .

[14]  Andreas Blass.  On Certain Types and Models for Arithmetic , 1974, J. Symb. Log..

[15]  John S. Schlipf,et al.  Toward model theory through recursive saturation , 1978, Journal of Symbolic Logic.

[16]  Ali Enayat.  On certain elementary extensions of models of set theory , 1984 .

[17]  R. Phillips.  A minimal extension that is not conservative. , 1974 .