Hypothesis: HR 71-alpha:
\(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119.
Conclusion: HR 193:
\(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2(\aleph_{\alpha})\) Jech's Model | This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\) |
| \(\cal N12(\aleph_{\alpha})\) A generalization of \(\cal N12(\aleph_1)\).Replace ``\(\aleph_1\)'' by ``\(\aleph_{\alpha}\)'' where \(\aleph_{\alpha}\) isa singular cardinal | Thus, \(|A|=\aleph_{\alpha}\); \(\cal G\) is the groupof all permutations on \(A\); and \(S\) is the set of all subsets of \(A\) withcardinality less than \(\aleph_{\alpha}\) |
| \(\cal N16(\aleph_{\gamma})\) Levy's Model II | This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\) |
| \(\cal N21(\aleph_{\alpha+1})\) Jensen's Model | We assume \(\aleph_{\alpha+1}\) is a regular cardinal |
Code: 5
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