Cohen \(\cal M20\): Felgner's Model I | Back to this models page
Description: Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\)
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. |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 231 | \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable. |
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 |
|---|---|
| 31 | \(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 131 | \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 144 | Every set is almost well orderable. |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 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: He then proves that in the resulting model, \(\calM20\), every set can be linearly ordered (30 is true) and the union of awell ordered set of well orderable sets can be well ordered (231 is true),but there is a denumerable set of denumerable sets which does not have adenumerable union (31 is false). Also, in this model every set of wellorderable sets has a choice function (60 is true). Form 60 implies 294(Every linearly ordered \(W\)-set is well orderable.), and 30 \(+\) 294 \(+\)144 (Every set is a \(W\)-set.) implies AC. It follows that 144 is false.(See Note 25 for definitions.) SinceForm 64 (There is no amorphous set.)is true (30 implies 64),Form 133 (Every set is either well orderable orhas an infinite amorphous subset.) is false.Form 40 (\(C(WO,\infty)\)) isfalse because 40 implies 31 andForm 122 (\(C(WO,<\aleph_0)\)) is truebecause 30 implies 122. Therefore,Form 328 (\(MC(WO,\infty)\)) is falsebecause \(122 + 328 \to 40\).
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