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Inverse limit reflection and the structure of \(L(V_{{\lambda}+1})\). (English) Zbl 1371.03070

Summary: We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding \(L_\alpha (V_{\lambda+1}) \to L_{\alpha}(V_{\lambda+1})\). Using these inverse limit reflection embeddings directly and by broadening the collection of \(U(j)\)-representable sets, we prove structural results of \(L(V_{\lambda+1})\) under the assumption that there exists an elementary embedding \(j : L(V_{\lambda+1}) \to L(V_{\lambda+1})\). As a consequence we show the impossibility of a generalized inverse limit \(X\)-reflection result for \(X \subseteq V_{\lambda+1}\), thus focusing the study of \(L(\mathbb{R})\) generalizations on \(L(V_{\lambda+1})\).

MSC:

03E55 Large cardinals
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[1] D. Gale and F. M. Stewart, Contributions to the Theory of Games, Annals of Mathematics Studies 2 (Princeton University Press, Princeton, NJ, 1953) pp. 245–266.
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[3] DOI: 10.1007/978-1-4020-5764-9_24 · Zbl 1198.03072 · doi:10.1007/978-1-4020-5764-9_24
[4] DOI: 10.1016/S0168-0072(97)00031-6 · Zbl 0890.03027 · doi:10.1016/S0168-0072(97)00031-6
[5] DOI: 10.1016/S0168-0072(00)00035-X · Zbl 0968.03060 · doi:10.1016/S0168-0072(00)00035-X
[6] DOI: 10.1007/BFb0071699 · doi:10.1007/BFb0071699
[7] DOI: 10.1142/S021906131100102X · Zbl 1248.03069 · doi:10.1142/S021906131100102X
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