In this paper we prove the following theorem:
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[{\rm {Con}}({\rm {ZFC}}\,{\rm { + }}\,there\,is\,a\,{\rm {2 - }}huge\,cardinal) \Rightarrow for\,all\,n
\[
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ℵ
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3
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↠
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{\rm {Con}}({\rm {ZFC + }}({\aleph _{n + 3}},{\aleph _{n + 2}},{\aleph _{n + 1}}) \twoheadrightarrow ({\aleph _{n + 2}},{\aleph _{n + 1}},{\aleph _n}))
\]
. We do this by using iterated forcing to collapse the
2
2
-huge cardinal to
ℵ
n
+
1
{\aleph _{n + 1}}
and extending the elementary embedding generically.