Cohen \(\cal M46(m,M)\): Pincus' Model VIII | Back to this models page
Description: This model depends on the natural number \(m\) and the set of natural numbers \(M\) which must satisfy Mostowski's condition:
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 |
|---|---|
| 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. |
| 46-K | If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function. |
| 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 \(|f(y)|=\aleph_{0}\). |
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 |
|---|---|
| 45-n | If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 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. |
| 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: (See Mostowski [1945] and Note 15.)\parindent=20pt\smallskipIt is the model of Pincus [1977a], Theorem 2.1\. (D) and usesPincus' general method for adding dependent choice (43). Beginning with amodel \(\cal M\) of ZF + ``The class form of choice'' Pincus adds threegeneric \(\omega_1\) sequences: \(\langle I_\alpha : \alpha < \omega_1\rangle\), \(\langle A_\alpha : \alpha < \omega_1 \rangle\) and \(\langle\frak A_\alpha : \alpha < \omega_1 \rangle\). The sets \(I_\alpha\), \(\alpha< \omega_1\) are obtained as in \(\cal M1(\langle\omega_1\rangle)\),\(A_\alpha\) is a generic set of disjoint subsets of \(I_\alpha\) and \(\frakA_\alpha\) is a generically added copy of the structure introduced inPincus [1974b] with domain \(A_\alpha\). (See Pincus [1977a] for definitions.) In Pincus [1977a], it is shownthat in \(\cal M46(m,M)\) Dependent Choice (43) is true, ``For everycollection \(A\) of sets each 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, ``Every set ofsets, all of whose cardinalities are in \(M\), has a choice function.''(46(\(M\))) is true but \(C(\infty,n)\) (45(\(n\))) is false.
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