Hypothesis: HR 60:
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.
Moore, G. [1982], p 125.
Conclusion: HR 384:
Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M3\) Mathias' model | Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\) |
| \(\cal M40(\kappa)\) Pincus' Model IV | The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\) |
| \(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) | \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\) |
| \(\cal N7\) L\"auchli's Model I | \(A\) is countably infinite |
Code: 3
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