Hypothesis: HR 305:
There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.
Conclusion: HR 126:
\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N1\) The Basic Fraenkel Model | The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\) |
| \(\cal N3\) Mostowski's Linearly Ordered 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 finitesubsets of \(A\) |
| \(\cal N4\) The Mathias/Pincus Model I | \(A\) is countably infinite;\(\precsim\) is a universal homogeneous partial ordering on \(A\) (See<a href="/articles/Jech-1973b">Jech [1973b]</a> p 101 for definitions.); \(\cal G\) is the group ofall order automorphisms on \((A,\precsim)\); and \(S\) is the set of allfinite subsets of \(A\) |
| \(\cal N5\) 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 \(A\) |
| \(\cal N17\) Brunner/Howard Model II | \(A=\{a_{\alpha,i}:\alpha\in\omega_1\,\wedge i\in\omega\}\) |
| \(\cal N18\) Howard's Model I | Let \(B= {B_n: n\in\omega}\) where the \(B_n\)'sare pairwise disjoint and each is countably infinite and let \(A=\bigcup B\) |
| \(\cal N24\) Hickman's Model I | This model is a variation of \(\cal N2\) |
| \(\cal N24(n)\) An extension of \(\cal N24\) to \(n\)-element sets, \(n>1\).\(A=\bigcup B\), where \( B=\{b_i: i\in\omega\}\) is a pairwise disjoint setof \(n\)-element sets | \(\cal G\) is the group of all permutations of \(A\)which are permutations of \(B\); and \(S\) is the set of all finite subsets of\(A\) |
| \(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\) | The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\) |
| \(\cal N29\) Dawson/Howard Model | Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\) |
| \(\cal N39\) Howard's Model II | \(A\) is denumerable and is a disjoint union\(\bigcup_{i\in\omega}B_i\cup\bigcup_{i\in\omega}C_i\), where for all\(i\in\omega, |B_i|=|C_i|=\aleph_0\) |
| \(\cal N41\) Another variation of \(\cal N3\) | \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\) |
| \(\cal N43\) Brunner's Model II | The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint |
| \(\cal N46\) Höft/Howard Model I | The set \(A\) of atoms is denumerable andordered by \(\le\) so that \((A,\le)\) is order isomorphic to the rationals.\(A\) is written as the union \(A = D_1 \cup D_2 \cup D_3\) of three densedisjoint subsets |
| \(\cal N48\) Pincus' Model XI | \(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\) |
| \(\cal N55\) Keremedis/Tachtsis Model: The set of atoms \(A=\bigcup \{A_n: n\in \omega\}\), where \(A_n=\{a_{n,x}: x\in B(0,\frac1n)\}\) and \(B(0,\frac1n)= \{x: \rho(x,0)=\frac1n\}\), where \(\rho\) is the Euclidean metric | The group of permutations \(\cal G\), is the group of all rotations of the \(A_n\) through an angle \(\theta\in [0,2\pi)\), and supports are finite |
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