This non-implication,
Form 23 \( \not \Rightarrow \)
Form 99,
whose code is 6,
is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 23 | <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </p> |
| Conclusion | Statement |
|---|---|
| Form 99 | <p> <strong>Rado's Selection Lemma:</strong> Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. </p> |
The conclusion Form 23 \( \not \Rightarrow \) Form 99 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N5\) 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 |