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A theorem on the isomorphism property

  • Renling Jin (a1)

Abstract

An -structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in are internal. A nonstandard universe is said to satisfy the κ-isomorphism property if for any two internally presented -structures and , where has less than κ many symbols, is elementarily equivalent to implies that is isomorphic to . In this paper we prove that the ℵ1-isomorphism property is equivalent to the ℵ0-isomorphism property plus ℵ1-saturation.

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[CK]Chang, Chen Chung and Keisler, H. Jerome, Model theory, North-Holland, Amsterdam, 1973; 3rd ed., 1990.
[H1]Henson, C. Ward, The isomorphism property in nonstandard analysis and its use in the theory of Banach space, this Journal, vol. 39 (1974), pp. 717731.
[H2]Henson, C. Ward, When do two Banach spaces have isometrically isomorphic nonstandard hulls, Israel Journal of Mathematics, vol. 22 (1975), pp. 5767.
[H3]Henson, C. Ward, Nonstandard hulls of Banach spaces, Israel Journal of Mathematics, vol. 25 (1976), pp. 108144.
[H4]Henson, C. Ward, Private communication.
[J]Jin, Renling, The isomorphism property versus the special model axiom, this Journal, vol. 57 (1992), (to appear).
[R]Ross, David, “The special model axiom in nonstandard analysis, this Journal, vol. 55 (1990), pp. 12331242.

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