Cohen \(\cal M13\): Feferman/Solovay Model | Back to this models page
Description: This model is an extension of \(\cal M2\) in which there are \(\omega_1\) generic real numbers, but no set to collect them
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 |
|---|---|
| 40 | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 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. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
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 |
|---|---|
| 44 | \(DC(\aleph _{1})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in X\) with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\). |
| 67 | \(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 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.) |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
Historical background: Solovay has shown that in this model, the Axiom of Choicefor a well ordered family of sets (40) and the Principle of DependentChoices (DC, 43) are true, while the extension of DC to \(\omega_1\) (44) isfalse. Since 40 implies 8 (\(C(\aleph_0,\infty)\)), it follows fromBrunner [1982a] that in this model there is a set that cannot bewell ordered and does not have an infinite Dedekind finite subset, (163 isfalse). (Form 8 plusForm 163 iff AC.)
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