Notes

Description:

Form 62 (\(C(\infty,< \aleph_0)\)) + Form 57 implies Form 9 is not provable in \(ZF\).

Content:

Sageev [1981] shows that the implication Form 62 (\(C(\infty,< \aleph_0)\)) + Form 57 (Any two Dedekind cardinals are comparable) implies Form 9 (Dedekind finite= finite) is not provable in \(ZF\).  Ellentuck [1974] shows that

\(ZF\) \(\vdash\) ( 62 + 57 \(\rightarrow  \langle\omega ,\cdot ,+\rangle\) is an elementary  submodel  of \(\langle\Delta ,\cdot ,+\rangle \,)\)
where \(\Delta \) is the set of Dedekind finite cardinals \((\omega\subseteq\Delta )\) and \(\cdot \)  and \(+\) are the usual cardinal operations.

Howard-Rubin number: 44

Type: Summary

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