Cohen \(\cal M1(\langle\omega_1\rangle)\): Cohen/Pincus Model | Back to this models page
Description: 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 \(\{I_{\alpha}:\alpha \le \lambda\}\)
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. |
| 97 | Cardinal Representatives: For every set \(A\) there is a function \(c\) with domain \({\cal P}(A)\) such that for all \(x, y\in {\cal P}(A)\), (i) \(c(x) = c(y) \leftrightarrow x\approx y\) and (ii) \(c(x)\approx x\). Jech [1973b] p 154. |
| 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 |
|---|---|
| 20 | If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in S\}\) are families of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8). |
| 39 | \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202. |
| 85 | \(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13. |
| 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. |
| 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:
Pincus proves that the Principle of Dependent Choices(43) is true, for each family of infinite sets \(x\), there is a function\(f\) such that for each \(y\in x\), \(f(y)\subseteq y\) and \(|f(y)|=\aleph_0\)(214), is true, every set can be linearly ordered (30) is true, there is aclass of cardinal representatives, henceForm 97 is true, but\(C(\aleph_1,\aleph_0)\) is false because \(