Axiom Of Choice

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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