Iterated extended ultrapowers and supercompactness without choice

Annals of Pure and Applied Logic 54 (2):179-194 (1991)

Abstract
Working in ZF + DC with no additional use of the axiom of choice, we show how to iterate the extended ultrapower construction of Spector . This generalizes the technique of iterated ultrapowers to choiceless set theory. As an application, we prove the following theorem: Assume V = LU[κ] + “κ is λ-supercompact with normal ultrafilter U” + DC. Then for every sufficiently large regular cardinal ρ, there exists a set-generic extension V[G] of the universe in which there exists for some σ a set S ρ for which one can define an elementary embedding j mapping V to LD[S], where D is the filter in V[G] generated by the closed unbounded filter on ρ. Moreover, we have j = ρ, j = σ, j) = S according to LD[S]), and j = D ∩ LD[S] i s a normal ultrafilter in LD[S] on ρ
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DOI 10.1016/0168-0072(91)90030-p
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References found in this work BETA

Strong Axioms of Infinity and Elementary Embeddings.Robert M. Solovay - 1978 - Annals of Pure and Applied Logic 13 (1):73.
Some Applications of Iterated Ultrapowers in Set Theory.Kenneth Kunen - 1970 - Annals of Pure and Applied Logic 1 (2):179.
Some Combinatorial Problems Concerning Uncountable Cardinals.Thomas J. Jech - 1973 - Annals of Pure and Applied Logic 5 (3):165.
On the Determinacy of Games on Ordinals.L. A. Harrington - 1981 - Annals of Pure and Applied Logic 20 (2):109.
Ultrapowers Without the Axiom of Choice.Mitchell Spector - 1988 - Journal of Symbolic Logic 53 (4):1208-1219.

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