Hypothesis: HR 337:
\(C(WO\), uniformly linearly ordered): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).
Conclusion: HR 163:
Every non-well-orderable set has an infinite, Dedekind finite subset.
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 |
| \(\cal M13\) 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 |
| \(\cal N2\) The Second Fraenkel Model | The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \). |
| \(\cal N12(\aleph_1)\) 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 \(A\) |
Code: 5
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