Hypothesis: HR 31:
\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
Conclusion: HR 328:
\(MC(WO,\infty)\): For every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M1\) Cohen's original model | Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them |
| \(\cal M1(\langle\omega_1\rangle)\) Cohen/Pincus Model | Pincus extends the methods of Cohen and adds a generic \(\omega_1\)-sequence, \(\langle I_{\alpha}: \alpha\in\omega_1\rangle\), of denumerable sets, where \(I_0\) is a denumerable set of generic reals, each \(I_{\alpha+1}\) is a generic set of enumerations of \(I_{\alpha}\), and for a limit ordinal \(\lambda\),\(I_{\lambda}\) is a generic set of choice functions for \(\{I_{\alpha}:\alpha \le \lambda\}\) |
| \(\cal M43\) Pincus' Model V | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((A)\) |
| \(\cal M44\) Pincus' Model VI | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((B)\) |
| \(\cal M45\) Pincus' Model VII | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\) |
| \(\cal N1\) 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 \(A\) |
| \(\cal N5\) 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 |
| \(\cal N15\) Brunner/Howard Model I | \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\) |
| \(\cal N17\) Brunner/Howard Model II | \(A=\{a_{\alpha,i}:\alpha\in\omega_1\,\wedge i\in\omega\}\) |
| \(\cal N19(\precsim)\) Tsukada's Model | Let \((P,\precsim)\) be a partiallyordered set that is not well ordered; Let \(Q\) be a countably infinite set,disjoint from \(P\); and let \(I=P\cup Q\) |
| \(\cal N24\) 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 \(\le_n\) |
| \(\cal N38\) 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 |
| \(\cal N39\) 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\(i\in\omega, |B_i|=|C_i|=\aleph_0\) |
| \(\cal N40\) Howard/Rubin Model II | A variation of \(\cal N38\) |
| \(\cal N48\) 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 \(a\in C_n\) |
Code: 3
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