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Isomorphic Ultrapowers

Henson, C. Ward and Iovino, José and Kechris, Alexander S. and Odell, Edward (2003) Isomorphic Ultrapowers. In: Analysis and Logic. London Mathematical Society Lecture Note Series. No.262. Cambridge University Press , Cambridge, pp. 55-71. ISBN 9781107360006.

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In this chapter we prove isomorphism theorems for ultrapowers and ultraproducts of normed space structures. These results show that there is a very tight connection between (a) properties that are preserved under the ultraproduct construction and (b) properties that are expressible using the logic for normed space structures that is described in this paper. Let L be a signature and let ℳ and N be two normed space L-structures. If ℳ and N have isomorphic ultrapowers, by Corollary 9.4 they must be approximately elementarily equivalent. Theorem 10.7 below gives the converse (in a strong form). Together these results show that ultrapower equivalence of ℳ and N is the same as approximate elementary equivalence. (See the discussion of this issue in the Introduction.) Theorem 10.8 below is a similar result for ultraproducts. Among other things, it shows that the ultrafilter guaranteed by Theorem 10.7 can be chosen in a highly uniform way and that the ultrapowers in question can be taken to be highly saturated. The uniformity will be exploited in Chapter 12 to prove the existence of ultrapowers that are highly homogeneous (in addition to being highly saturated). The results in this chapter are analogous to the Keisler-Shelah Theorem in ordinary model theory. (See [She71] and Chapter 6 in [CK90].) Moreover, our proof follows a similar line of argument, with adjustments appropriate to the handling of positive bounded formulas and their approximations.

Item Type:Book Section
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Additional Information:© 2003 Cambridge University Press.
Series Name:London Mathematical Society Lecture Note Series
Issue or Number:262
Record Number:CaltechAUTHORS:20180816-160145747
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Official Citation:Henson, C., Iovino, J., Kechris, A., & Odell, E. (2003). Isomorphic Ultrapowers. In C. Finet & C. Michaux (Eds.), Analysis and Logic (London Mathematical Society Lecture Note Series, pp. 55-71). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107360006.012
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88886
Deposited By: George Porter
Deposited On:17 Aug 2018 14:32
Last Modified:16 Nov 2021 00:30

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