This non-implication,
Form 423 \( \not \Rightarrow \)
Form 163,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 165 | <p> \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. </p> |
| Conclusion | Statement |
|---|---|
| Form 163 | <p> Every non-well-orderable set has an infinite, Dedekind finite subset. </p> |
The conclusion Form 423 \( \not \Rightarrow \) Form 163 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M2\) Feferman's model | Add a denumerable number of generic reals to the base model, but do not collect them |
| \(\cal M11\) Forti/Honsell Model | Using a model of \(ZF + V = L\) for the ground model, the authors construct a generic extension, \(\cal M\), using Easton forcing which adds \(\kappa\) generic subsets to each regular cardinal \(\kappa\) |
| \(\cal M13\) Feferman/Solovay Model | This model is an extension of <a href="/models/Feferman-1">\(\cal M2\)</a> in which there are \(\omega_1\) generic real numbers, but no set to collect them |
| \(\cal M25\) Freyd's Model | Using topos-theoretic methods due to Fourman, Freyd constructs a Boolean-valued model of \(ZF\) in which every well ordered family of sets has a choice function (<a href="/form-classes/howard-rubin-40">Form 40</a> is true), but \(C(|\Bbb R|,\infty)\) (<a href="/form-classes/howard-rubin-181">Form 181</a>) is false |
| \(\cal M40(\kappa)\) Pincus' Model IV | The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\) |
| \(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) | Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false |
| \(\cal N12(\aleph_2)\) Another variation of \(\cal N1\) | Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above |
| \(\cal N33\) Howard/H\.Rubin/J\.Rubin Model | \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all boundedsubsets of \(A\) |
| \(\cal N38\) Howard/Rubin Model I | Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering |
| \(\cal N40\) Howard/Rubin Model II | A variation of \(\cal N38\) |