Cohen \(\cal M34(\aleph_1)\): Pincus' Model III | Back to this models page
Description: Pincus proves that Cohen's model \(\cal M1\) can be extended by adding \(\aleph_1\) generic sets along with the set \(b\) containing them and well orderings of all countable subsets of \(b\)
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 |
|---|---|
| 15 | \(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)). |
| 85 | \(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13. |
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 |
|---|---|
| 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. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 213 | \(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function. |
| 330 | \(MC(WO,WO)\): For every well ordered set \(X\) 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: The model is constructed in such a way that every collection ofdenumerable sets has a choice function (85 is true), but there is adenumerable collection of sets of cardinality \(\aleph_1\) which does nothave a choice function. Therefore, both \(C(WO,WO)\) (165) and\(C(\infty,\aleph_1)\) (213) are false. Pincus also shows that theKinna-Wagner Principle (15) and the Ordering Principle (30) are true.SinceForm 64 (There is no amorphous set.) is true (30 implies 64), form133 (Every set is either well orderable or has an infinite amorphoussubset.) is false.Form 165 is false andForm 122 (\(C(WO,<\aleph_0)\)) istrue because 15 implies 122. Therefore,Form 330 (\(MC(WO,WO)\)) is falsebecause \(122 + 330 \to 165\).
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