This non-implication,
Form 31 \( \not \Rightarrow \)
Form 181,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 31 | <p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong> The union of a denumerable set of denumerable sets is denumerable. </p> |
| Conclusion | Statement |
|---|---|
| Form 418 | <p> DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable. </p> |
The conclusion Form 31 \( \not \Rightarrow \) Form 181 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N60\) de la Cruz-Hall model 3 | Let \(\{ R_{n,i} : n, i \in \omega \}\) be a partition of \(A\) (the set of atoms) into continuum sized sets and fix bijections \(f_{n,i} : \mathbb R \to R_{n.i}\). For each \(n \in \omega\) let \(A_{n} = \bigcup \{ R_{n,i} : i \in \omega \}\) and let \(D_{n}\) be the metric on \(A_{n}\) such that each \(f_{n,i}\) is an isometry and the distance betweenn elements in different \(R_{n,i}\)'s is always \(1\). Let \(G^{+}\) be the group of permutations \(\pi\) of A such that <ol type="a"> <li>\(\pi\) is the identity on all but finitely many of the \(R_{n,i}\)'s.</li> <li>For all \(n\) and \(i\) in \(\omega\), there is a \(j \in \omega\) such that \(\pi(R_{n,i}) = R_{n,j}\) (so that \(\pi(A_{n}) = A_{n}\)).</li> <li>If \(\pi(R_{n,i}) = R_{n,j}\) then \(f^{-1}_{n,j} \circ \pi \circ f_{n,i}\) is an affine transformation of \(\mathbb R\).</li> </ol> Let \(\mathcal D = \{ \pi(D_{n}) : n \in \omega \mbox{ and } \pi \in G^{+} \}\). The group \(G\) used to define the permutation model is the group generated by elements \(g \in G^{+}\) such that \(g\) is an isometry of some \(D \in \mathcal{D}\). Let \(\mathcal{W}\) be the set of relations \(w\) such that \(w\) is a well-ordering of some \(R_{n,i}\). The set of supports is the set of finite subsets of \(\mathcal{D} \cup \mathcal{W}\). |