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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: $[{\rm {Con}}({\rm {ZFC}} {\rm { + }} there is a {\rm {2 - }}huge cardinal) \Rightarrow for all n$ \[ {\rm {Con}}({\rm {ZFC + }}({\aleph _{n + 3}},{\aleph _{n + 2}},{\aleph _{n + 1}}) \twoheadrightarrow ({\aleph _{n + 2}},{\aleph _{n + 1}},{\aleph _n}))\]. 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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Keywords: Chang’s Conjecture, large cardinals, iterated forcing, master conditions
Article copyright: © Copyright 1982 American Mathematical Society