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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Large cardinals and strong model theoretic transfer properties


Author: Matthew Foreman
Journal: Trans. Amer. Math. Soc. 272 (1982), 427-463
MSC: Primary 03C55; Secondary 03E35, 03E55
DOI: https://doi.org/10.1090/S0002-9947-1982-0662045-X
MathSciNet review: 662045
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Abstract: In this paper we prove the following theorem: $ [{\rm {Con}}({\rm {ZFC}}\,{\rm { + }}\,there\,is\,a\,{\rm {2 - }}huge\,cardinal) \Rightarrow for\,all\,n$

$\displaystyle {\rm {Con}}({\rm {ZFC + }}({\aleph _{n + 3}},{\aleph _{n + 2}},{\... ...{n + 1}}) \twoheadrightarrow ({\aleph _{n + 2}},{\aleph _{n + 1}},{\aleph _n}))$

. We do this by using iterated forcing to collapse the $ 2$-huge cardinal to $ {\aleph _{n + 1}}$ and extending the elementary embedding generically.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0662045-X
Keywords: Chang's Conjecture, large cardinals, iterated forcing, master conditions
Article copyright: © Copyright 1982 American Mathematical Society