Cohen \(\cal M32\): Sageev's Model II | Back to this models page
Description: Starting with a model \(\cal M\) of \(ZF + V =L\), Sageev constructs a sequence of models \(\cal M\subseteq N_0 \subseteq N_1\subseteq\cdots\subseteq N_{\kappa}\) where \(\kappa\) is an inaccessible cardinal, \(N_0\) is Cohen's model \(\cal M1\), and \(N_{\kappa}\) is \(\cal M32\)
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. |
| 57 |
If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\). |
| 62 | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite 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 |
|---|---|
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 126 | \(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). |
| 133 | Every set is either well orderable or has an infinite amorphous 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.) |
Historical background: In this model, every set can be linearly ordered (30 is true); if\(\Delta\) is the set of Dedekind finite cardinals, then\(2^{\aleph_0}<|\Delta|\le 2^{2^{\aleph_0}}\) (9 is false) and each pair ofcardinals in \(\Delta\) is comparable (57 is true). Sageev also shows that\(C(\infty, <\aleph_0)\) (62) is true. SinceForm 64 (There is noamorphous set.) is true (30 implies 64),Form 133 (Every set is eitherwell orderable or has an infinite amorphous subset.) is false.Form 40(\(C(WO,\infty)\)) is false because 40 implies 9 andForm 122(\(C(WO,<\aleph_0)\)) is true because 30 implies 122. Therefore,Form 328(\(MC(WO,\infty)\)) is false because \(122 + 328 \to 40\). In addition, 126(\(MC(\aleph_0,\infty)\)) is false because 30 is true and 8 is false. (8(\(C(\aleph_0,\infty)\)) implies 9 and 30 + 126 implies 8).
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