Cohen \(\cal M45\): Pincus' Model VII | Back to this models page
Description: This is the model of Pincus [1977a], Theorem 2.1 \((C)\)
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 |
|---|---|
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 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. |
| 214 | \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). |
| 295 | DO: Every infinite set has a dense linear ordering. |
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 |
|---|---|
| 49 | Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78. |
| 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. |
| 99 | Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 151 | \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).) |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 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.) |
| 329 | \(MC(\infty,WO)\): For every set \(M\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.) |
Historical background: (C). It uses Pincus' general method foradding dependent choice (43). Beginning with a model \(\cal M\) of ZF +``The class form of choice'', Pincus adds three generic \(\omega_1\)sequences: \(\langle I_\alpha : \alpha < \omega_1 \rangle\), \(\langleA_\alpha : \alpha < \omega_1 \rangle\) and \(\langle \frak A_\alpha : \alpha< \omega_1 \rangle\). The sets \(I_\alpha\), \(\alpha < \omega_1\) areobtained as in \(\cal M1(\langle\omega_1\rangle)\), \(A_\alpha\) is a genericset of disjoint subsets of \(I_\alpha\) and \(\frak A_\alpha\) is agenerically added universal homogeneous structure for the theory ofpartial orders together with a dense linear order both with domain\(A_\alpha\). (See Pincus [1977a] for definitions.) In Pincus [1977a], it is shown that in \(\cal M45\) the Ordering Principle (30)is true, Dependent Choice (43) is true, ``For every collection \(A\) of setseach of which has cardinality at least \(\aleph_0\), there is a function \(f\)with domain \(A\) such that \(\forall x\in A\), \(f(x)\subseteq x\) and \(|f(x)|= \aleph_0\).'' (214) is true but the Order Extension Principle (49) isfalse.Form 43 impliesForm 296 (Every infinite set is the disjoint unionof infinitely many infinite sets.) and it is shown in Pincus [1997] that 30 + 296 implies 295 (Every infinite set has a denselinear ordering.) Thus,Form 295 is true.
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