Annals of Pure and Applied Logic 149 (1-3):1-6 (2007)

Abstract
The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:Vλ→Vλ, the existence of such a j which is moreover , and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown : if V is a model of ZFC and V[G] is a -generic forcing extension of V, then in V[G], V is definable using the parameter Vδ+1, where
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DOI 10.1016/j.apal.2007.07.002
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References found in this work BETA

Elementary Embeddings and Infinitary Combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
Implications Between Strong Large Cardinal Axioms.Richard Laver - 1997 - Annals of Pure and Applied Logic 90 (1-3):79-90.
Coding Lemmata in L.George Kafkoulis - 2004 - Archive for Mathematical Logic 43 (2):193-213.

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Citations of this work BETA

The Set-Theoretic Multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
Set-Theoretic Geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
The Downward Directed Grounds Hypothesis and Very Large Cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
Suitable Extender Models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
Universism and Extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - forthcoming - Review of Symbolic Logic:1-50.

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