Abstract
Let \({\fancyscript{L}}\) be a finite first-order language and \({\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}\) be a sequence of finite \({\fancyscript{L}}\)-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ω2 is non-empty, then there is a non-principal ultrafilter \({\fancyscript{U}}\) over ω such that the corresponding ultraproduct \({\prod_\fancyscript{U}\fancyscript{M}_n}\) has an automorphism that is not induced by an element of \({\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}\) .
References
Chang C.C., Keisler H.J.: Model Theory, 2nd edn, Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam (1977)
Kechris A.S.: Classical Descriptive Set Theory, vol 156 of Graduate Texts in Mathematics. Springer, New York, NY (1995)
Lücke P., Thomas S.: Automorphism groups of ultraproducts of finite symmetric groups. Commun. Algebra 39(10), 3625–3630 (2011)
Shelah S.: On the cardinality of ultraproduct of finite sets. J. Symb. Log. 35, 83–84 (1970)
Shelah S.: Vive la différence. III. Israel J. Math. 166, 61–96 (2008)
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Lücke, P., Shelah, S. External automorphisms of ultraproducts of finite models. Arch. Math. Logic 51, 433–441 (2012). https://doi.org/10.1007/s00153-012-0271-1
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DOI: https://doi.org/10.1007/s00153-012-0271-1