Hypothesis: HR 0: \(0 = 0\).
Conclusion: HR 66:
Every vector space over a field has a basis.
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 |
| \(\cal M2(\kappa)\) Feferman/Pincus Model | This is an extension of <a href="/models/Feferman-1">\(\cal M2\)</a> in which there are \(\kappa\) generic sets, where \(\kappa\) is a regular cardinal |
| \(\cal M2(\langle\omega_2\rangle)\) Feferman/Truss Model | This is another extension of <a href="/models/Feferman-1">\(\cal M2\)</a> |
| \(\cal M3\) Mathias' model | Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\) |
| \(\cal M4\) Pincus' Model I | This model has many of the properties of the model given in <a href="/books/39">Cohen [1966]</a> (p 143) |
| \(\cal M5(\aleph)\) 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 \(\Bbb R\) |
| \(\cal M6\) Sageev's Model I | Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\) |
| \(\cal M7\) Cohen's Second Model | There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model |
| \(\cal M7(n)\) Generalization of \(\cal M7\) | Model <a href="/models/Cohen-2">\(\cal M7\)</a> can be generalized to \(n\) denumerable sets for \(1 \le n \in\omega\), then the Axiom of Choice for a denumerable number of \(n\) element sets, \(C(\aleph_0,n)\), is false for \(1 \le n \le \omega\) |
| \(\cal M8\) Apter's Model | Suppose \(\cal M \models ZFC +\) "There arecardinals \(\kappa \le \delta \le \lambda\) such that \(\kappa\) is a supercompact limit of supercompact cardinals, \(\lambda\) is a measurable cardinal, and \(\delta\) is \(\lambda\) supercompact." (See, for example, <a href="/articles/Drake-1974">Drake [1974]</a>, <a href="/articles/Kanamori-Magador-1978">Kanamori/Magador [1978]</a>, or <a href="/articles/Solovay-Reinhardt-Kanamori-1978">Solovay/Reinhardt/Kanamori [1978]</a> for information about large cardinals.) \(\cal M8\) is constructed by first forcing over the ground model \(\cal M\), constructing an inner model \(\cal M'\), doing an additional forcing argument over \(\cal M'\), and then constructing the final inner model \(\cal M8\) |
| \(\cal M9\) Feferman/Levy Model | Assume the ground model, \(\cal M\), satisfies \(ZF + GCH\) (the <strong>Generalized Continuum Hypothesis</strong>) |
| \(\cal M10\) Derrick/Drake Model | Let \(\cal M\) be a model of \(ZF + GCH\). Add to \(\cal M\) generic functions \(f_n\) for each \(n\in\omega\), where \(f_n:\omega_n\to\cal P(\omega)\), but do not add \(\{f_n: n\in\omega\}\) |
| \(\cal M11\) 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 \(\kappa\) |
| \(\cal M12(\aleph)\) Truss' Model I | This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular |
| \(\cal M13\) 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 |
| \(\cal M15\) Feferman/Blass Model | Blass constructs a model similar to Feferman's model, <a href="/models/Feferman-1">\(\cal M2\)</a> |
| \(\cal M16(n)\) Monro's Model I | Monro has constructed an \(\omega\)-sequence of models of \(ZF\) such that, <ul type="none"> <li>(*) For each ordinal \(\alpha\), \(\cal P^n(\alpha)\) is the same in the \(n\)th model and the \(n+1\)st model.</li> </ul> (\(\cal P^0(\alpha)=\alpha,\) and \(\cal P^{n+1}(\alpha)=\cal P(\cal P^n(\alpha))\).) He starts with a countable transitive model \(\cal M\models ZF + V = L\) |
| \(\cal M17\) Gitik's Model | Using the assumption that for every ordinal \(\alpha\) there is a strongly compact cardinal \(\kappa\) such that \(\kappa >\alpha\), Gitik extends the universe \(V\) by a filter \(G\) generic over a proper class of forcing conditions |
| \(\cal M18\) 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) |
| \(\cal M19(\aleph)\) Monro's Model II | Let \(\cal M\) be a countable transitive model of \(ZF + V = L\) and let \(\aleph\) be a regular cardinal in \(\cal M\) |
| \(\cal M20\) Felgner's Model I | Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\) |
| \(\cal M21\) Felgner's Model II | Suppose \(\cal M \models ZF + V = L\). Define \(B=\{f: (\exists\alpha <\omega_1)f:\alpha\to\omega\}\) |
| \(\cal M22\) Plotkin's Model I | Let \(T\) be a complete first order theory with equality which has infinite models and is \(\aleph_0\)-categorical |
| \(\cal M23\) Hodges' Model | Let \(\cal M\) be a countable transitive model of \(ZFC + V = L\) and let \(\kappa\) be a regular cardinal in \(\cal M\) |
| \(\cal M24\) Blass' Model | Let \(\cal M\) be a countable transitive model of \(ZFC + V = L\) |
| \(\cal M25\) 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 |
| \(\cal M26\) Kanovei's Model I | Starting with a model of \(ZF + V = L\) and using forcing techniques due to <a href="/excerpts/Jensen-1968">Jensen [1968]</a>, Kanovei constructs a model of \(ZF\) in which there is an infinite Dedekind finite set \(A\) of generic reals that is in the class \(\varPi^1_n\), but there are no infinite Dedekind finite subsets of \(\Bbb R\) in the class \(\varSigma^1_n\), where \(n\in\omega\), \(n\ge 2\) |
| \(\cal M27\) Pincus/Solovay Model I | Let \(\cal M_1\) be a model of \(ZFC + V =L\) |
| \(\cal M28\) Morris' Model II | Morris constructs a generic extension of acountable standard model of ZFC in which there is a proper class ofgeneric sets |
| \(\cal M29\) Pincus' Model II | Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\) |
| \(\cal M30\) 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) |
| \(\cal M31\) Szczepaniak's Model | Two models of \(ZF\), \(M_1\) and \(M_2\),\(M_1\subseteq M_2\), are constructed with the same ordinals (so \(L^1\) in\(M_1\) is the same as \(L^1\) in \(M_2\)), and a generic real \(a\in M_1\) so that \(a\not\in HOD\) in \(M_1\), but \(a\in HOD\) in \(M_2\) |
| \(\cal M32\) Sageev's Model II | Starting with a model \(\cal M\) of \(ZF + V =L\), Sageev constructs a sequence of models \(\cal M\subseteq N_0 \subseteq N_1\subseteq\cdots\subseteq N_{\kappa}\) where \(\kappa\) is an inaccessible cardinal, \(N_0\) is Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a>, and \(N_{\kappa}\) is \(\cal M32\) |
| \(\cal M33\) Plotkin's Model II | The construction is similar to the construction of <a href="/models/Plotkin-1">\(\cal M22\)</a> |
| \(\cal M34(\aleph_1)\) Pincus' Model III | Pincus proves that Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a> can be extended by adding \(\aleph_1\) generic sets along with the set \(b\) containing them and well orderings of all countable subsets of \(b\) |
| \(\cal M35(\epsilon)\) David's Model | In Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a>, define sets \(B_n=\{x\subset\omega: |x\ \Delta\ a_n| <\omega\vee |x\ \Delta\ \omega-a_n| \le\omega\}\) (where \(\Delta\) is the symmetric difference) |
| \(\cal M36\) Figura's Model | Starting with a countable, standard model, \(\cal M\), of \(ZFC + 2^{\aleph_0}=\aleph_{\omega +1}\), Figura uses forcing conditions that are functions from a subset of \(\omega\times\omega\) to \(\omega_\omega\) to construct a symmetric extension of \(\cal M\) in which there is an uncountable well ordered subset of the reals (<a href="/form-classes/howard-rubin-170">Form 170</a> is true), but \(\aleph_1= \aleph_{\omega}\) so \(\aleph_1\) is singular (<a href="/form-classes/howard-rubin-34">Form 34</a> is false) |
| \(\cal M37\) Monro's Model III | This is a generic extension of <a href="/models/Cohen-1">\(\cal M1\)</a> in which there is an amorphous set (<a href="/form-classes/howard-rubin-64">Form 64</a> is false) and \(C(\infty,2)\) (<a href="/form-classes/howard-rubin-88">Form 88</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 \(AC\) |
| \(\cal M41\) Kanovei's Model III | Let \(\Bbb P\) be the set of conditions from the model in <a href="/excerpts/Jensen-1968">Jensen [1968]</a> |
| \(\cal M42\) 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 \(A\) |
| \(\cal N2\) 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 \( A \). |
| \(\cal N2(n)\) A generalization of \(\cal N2\) | This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\) |
| \(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) | \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\) |
| \(\cal N2(\aleph_{\alpha})\) Jech's Model | This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\) |
| \(\cal N2(\hbox{LO})\) van Douwen's Model | This model is another variationof \(\cal N2\) |
| \(\cal N3\) 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 \(A\) |
| \(\cal N7\) L\"auchli's Model I | \(A\) is countably infinite |
| \(\cal N8\) L\"auchli's Model II | \(A\) is countably infinite, ordered likethe rational numbers; \(\cal G\) is the group of all order automorphisms of\(A\); and \(S\) is the set of all subsets \(E\) of \(A\) such that \(E\) has atmost a finite number of accumulation points and every infinite subset of\(E\) has an accumulation point |
| \(\cal N10\) Höft/Howard/Mostowski Model | (The model is a variation of\(\cal N3\).) \(A\) as ordered by \(\precsim\) has the same order type as therationals; \(\cal G\) is the group of all order automorphisms of \(A\); \(S\) isthe set of all subsets \(E\) of \(A\) that satisfy the following threeconditions:\item{1.} \(E\) is well ordered by \(\precsim\).\item{2.} \(E\) is bounded in \(A\).\item{3.} If \(b:\alpha\to E\) is an order preserving bijection from\(\alpha\) onto \(E\) and if \(\lambda < \alpha\) is a limit ordinal then\(\{b(\beta) : \beta < \gamma\}\) has no least upper bound in\((A,\precsim)\).\par\noindentIn <a href="/articles/H\"oft/Howard-1994">H\"oft/Howard [1994]</a> it is shown that, in \(\cal N10\), everyDedekind finite set is finite (9 is true), but \((A,\precsim)\) is alinearly ordered set with no infinite descending sequences that cannot bewell ordered (77 is false) |
| \(\cal N12(\aleph_1)\) 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 |
| \(\cal N15\) Brunner/Howard Model I | \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\) |
| \(\cal N16\) Jech/Levy/Pincus Model | \(A\) has cardinality \(\aleph_{\omega}\);\(\cal G\) is the group of all permutations on \(A\); and \(S\) is the set ofall subsets of \(A\) of cardinality less that \(\aleph_{\omega}\) |
| \(\cal N16(\aleph_{\gamma})\) Levy's Model II | This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\) |
| \(\cal N17\) 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 \(A=\bigcup B\) |
| \(\cal N19(\precsim)\) 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 \(I=P\cup Q\) |
| \(\cal N21(\aleph_{\alpha+1})\) Jensen's Model | We assume \(\aleph_{\alpha+1}\) is a regular cardinal |
| \(\cal N22(p)\) Makowski/Wi\'sniewski/Mostowski Model | (Where \(p\) is aprime) Let \(A=\bigcup\{A_i: i\in\omega\}\) where The \(A_i\)'s are pairwisedisjoint and each has cardinality \(p\) |
| \(\cal N24\) 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\(A\) |
| \(\cal N24(n,LO)\) Truss' Model III | This is a variation of \(\cal N24(n)\)in which the set \(B\) is linearly ordered |
| \(\cal N26\) 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 \(\le_n\) |
| \(\cal N33\) 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 \(A\) |
| \(\cal N36(\beta)\) Brunner/Howard Model III | This model is a modificationof \(\cal N15\) |
| \(\cal N37\) A variation of Blass' model, \(\cal N28\) | Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\) |
| \(\cal N38\) 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 \(\cal N38\) |
| \(\cal N41\) 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 |
| \(\cal N47\) Höft/Howard Model II | This model is similar to \(\cal N33\).The atoms \(A\) are ordered by \(\le\) so that they have order type that ofthe real numbers \(\Bbb R\) (\(|A| = 2^{\aleph_0}\)) |
| \(\cal N48\) 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 \(a\in C_n\) |
| \(\cal N49\) De la Cruz/Di Prisco Model | Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\) |
| \(\cal N51\) Weglorz/Brunner Model | Let \(A\) be denumerable and \(\cal G\)be the group of all permutations of \(A\) |
| \(\cal N52\) Felgner/Truss Model | Let \((\cal B,\prec)\) be a countableuniversal homogeneous linearly ordered Boolean algebra, (i.e., \(<\) is alinear ordering extending the Boolean partial ordering on \(B\)) |
| \(\cal N53\) Good/Tree/Watson Model I | Let \(A=\bigcup \{Q_n:\ n\in\omega\}\), where \(Q_n=\{a_{n,q}:q\in \Bbb{Q}\}\) |
| \(\cal N54\) Good/Tree/Watson Model II | This model is a variation of \(\cal N 53\) |
| \(\cal N55\) 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 |
| \(\cal N56\) Howard's model III: Assume the the atoms are indexed asfollows: \(A = \{a(i,j) : i\in{\Bbb Q} \hbox{ and } j\in\omega \}\) | Foreach \(i\in \Bbb Q\), let \(A_i = \{a(i,j) : j\in \omega\}\) |
| \(\cal N57\) The set of atoms \(A=\cup\{A_{n}:n\in\aleph_{1}\}\), where\(A_{n}=\{a_{nx}:x\in B(0,1)\}\) and \(B(0,1)\) is the set of points on theunit circle centered at 0 | The group of permutations \(\cal{G}\) is thegroup of all permutations on \(A\) which rotate the \(A_{n}\)'s by an angle\(\theta_{n}\in\Bbb{R}\) and supports are countable |
| \(\cal N58\) Keremedis/Tachtsis Model 2: For each \(n\in\omega-\{0\}\), let\(A_n=\{({i\over n}) (\cos t,\sin t): t\in [0.2\pi)\}\) and let the set of atoms\(A=\bigcup \{A_n: n\in\omega-\{0\}\}\) | \(\cal G\) is the group of allpermutations on \(A\) which rotate the \(A_n\)'s by an angle \(\theta_n\), andsupports are finite |
| \(\cal M14\) Morris' Model I | This is an extension of Mathias' model, <a href="/models/Mathias-1">\(\cal M3\)</a> |
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