Cohen \(\cal M12(\aleph)\): Truss' Model I | Back to this models page
Description: This is a variation of Solovay's model, \(\cal M5(\aleph)\) in which \(\aleph\) is singular
When the book was first being written, only the following form classes were known to be true in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 6 | \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable. |
| 231 | \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable. |
| 369 | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
When the book was first being written, only the following form classes were known to be false in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 31 | \(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. |
| 34 | \(\aleph_{1}\) is regular. |
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 106 | Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire. |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 169 | There is an uncountable subset of \({\Bbb R}\) without a perfect subset. |
| 273 | There is a subset of \({\Bbb R}\) which is not Borel. |
| 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.) |
Historical background: (Truss uses themethod of collapsing cardinal which is due to A\. Levy, but, unlike \(\calM5\), \(\aleph\) is not an inaccessible cardinal.) Let \(\cal M\) be acountable model of ZF + V = L and let \(\aleph\) be a limit cardinal in\(\cal M\). For each ordinal \(\alpha\), the set of conditions \(Q^{\alpha}\) isthe set of finite sets \(p\) of triples, \((\beta,n,\gamma)\) where\(\gamma<\beta<\alpha\) and \(n\in\omega\) such that if \((\beta,n,\gamma_1),(\beta,n,\gamma_2)\in p\), then \(\gamma_1=\gamma_2\). \(Q_{\alpha}\) is theset of finite sets \(p\) of pairs of the form \((n,\gamma)\) where\(\gamma<\alpha\) and \(n\in\omega\) such that if \((n,\gamma_1),(n,\gamma_2)\in p\) then \(\gamma_1=\gamma_2\). A function \(f:\omega\to\alpha\) is called an \(\cal M\)-generic collapsing map if theset of finite subsets of \(\{(n,\gamma): f(n)=\gamma\}\) is an \(\calM\)-generic subset of \(Q_{\alpha}\). Let \(G\) be an \(\cal M\)-generic subsetof \(Q^{\aleph}\) then, \(G_\alpha\), the projection of \(G\) onto \(Q_{\alpha}\)is an \(\cal M\)-generic subset of \(Q_{\alpha}\), for each \(\alpha <\aleph\).Let \(f_{\alpha}= \bigcup G_{\alpha}\). Then \(\cal M12(\aleph)\) is thesmallest model of ZF containing the same ordinals as \(\cal M\) and each\(\langle f_{\beta}: \beta <\alpha\rangle\) for \(\alpha<\aleph\). In \(\calM12(\aleph)\), \(\aleph=\omega_1\). If \(\aleph\) is singular, \(\omega_1\) issingular so 34 is false. On the other hand, the union of a denumerable setof denumerable sets of reals is denumerable (6 is true) and the union of awell ordered set of well orderable sets can be well ordered (231 istrue). \(\cal P(\cal P(\omega))\) cannot be linearly ordered, so 30 isfalse. Truss also proves that the Principle of Dependent Choices (43) isfalse, every set of reals is Borel (273 is false), and every uncountableset of reals has a perfect subset (169 is false). It follows from [0 AB]that every perfect subset of \(\Bbb R\) has cardinality \(2^{\aleph_0}\).Thus, since 169 is false,Form 369 (If \(\Bbb R\) is partitioned into twosets, at least one of them has cardinality \(2^{\aleph_0}\).) is true. Form40 (\(C(WO,\infty)\)) is false because 40 implies 31 andForm 122(\(C(WO,<\aleph_0)\)) is true because 231 implies 122. Therefore,Form 328(\(MC(WO,\infty)\)) is false because \(122 + 328 \to 40\).
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