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 292:
\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).
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 \(\{I_{\alpha}:\alpha \le \lambda\}\) |
| \(\cal M2\) Feferman's model | Add a denumerable number of generic reals to the base model, but do not collect them |
| \(\cal M2(\langle\omega_2\rangle)\) Feferman/Truss Model | This is another extension of <a href="/models/Feferman-1">\(\cal M2\)</a> |
| \(\cal M5(\aleph)\) Solovay's Model | An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\) |
| \(\cal M13\) 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 |
| \(\cal M18\) Shelah's Model I | Shelah modified Solovay's model, <a href="/models/Solovay-1">\(\cal M5\)</a>, and constructed a model without using an inaccessible cardinal in which the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true and every set of reals has the property of Baire (<a href="/form-classes/howard-rubin-142">Form142</a> is false) |
| \(\cal M21\) Felgner's Model II | Suppose \(\cal M \models ZF + V = L\). Define \(B=\{f: (\exists\alpha <\omega_1)f:\alpha\to\omega\}\) |
| \(\cal M25\) 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 |
| \(\cal M27\) Pincus/Solovay Model I | Let \(\cal M_1\) be a model of \(ZFC + V =L\) |
| \(\cal M30\) Pincus/Solovay Model II | In this construction, an \(\omega_1\) sequence of generic reals is added to a model of \(ZFC\) in such a way that the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true, but no nonprincipal measure exists (<a href="/form-classes/howard-rubin-223">Form 223</a> is false) |
| \(\cal M38\) Shelah's Model II | In a model of \(ZFC +\) "\(\kappa\) is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse \(\kappa\) to \(\aleph_1\) similarly to <a href="/articles/Solovay-1970">Solovay [1970]</a> |
| \(\cal M39(\kappa,\lambda)\) Kanovei's model II | This model depends on the two cardinals \(\kappa < \lambda\) such that both \(\kappa\) and \(\lambda\) have cofinality \(>\omega\) and neither \(\kappa\) nor \(\lambda\) can be written as \(\theta^+\) where \(\theta\) is a cardinal of countable cofinality and such that \(\aleph_2 \le\kappa\) |
| \(\cal M40(\kappa)\) Pincus' Model IV | The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\) |
| \(\cal M42\) Bull's Model | Let \(\cal M\) be a countable transitive model of \(ZFC +\) "There are uncountable regular cardinals \(\aleph_\alpha <\aleph_\beta < \aleph_\gamma\) such that \(\aleph_\alpha\) is \(\aleph_\gamma\)-supercompact; \(\aleph_\beta\) is the first measurable cardinal greater than \(\aleph_\alpha\); and \(\aleph_\gamma =|2^{\aleph_\beta}|\)." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of \(\cal M\) |
| \(\cal M43\) 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 \((C)\) |
| \(\cal M46(m,M)\) Pincus' Model VIII | This model depends on the natural number \(m\) and the set of natural numbers \(M\) which must satisfy Mostowski's condition: <ul type="none"> <li>\(S(M,m)\): For everydecomposition \(m = p_{1} + \ldots + p_{s}\) of \(m\) into a sum of primes at least one \(p_{i}\) divides an element of \(M\)</li> </ul> |
| \(\cal M47(n,M)\) Pincus' Model IX | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\) |
| \(\cal N1\) 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 \(A\) |
| \(\cal N15\) Brunner/Howard Model I | \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\) |
| \(\cal N19(\precsim)\) 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 \(I=P\cup Q\) |
| \(\cal N29\) 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 |
| \(\cal N40\) 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 \(a\in C_n\) |
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