Cohen \(\cal M14\): Morris' Model I | Back to this models page
Description: This is an extension of Mathias' model, \(\cal M3\)
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 |
|---|---|
| 51 | Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117. |
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 14 | BPI: Every Boolean algebra has a prime ideal. |
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 |
|---|---|
| 49 | Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 99 | Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 144 | Every set is almost well orderable. |
Historical background: Morris shows that the cofinal subsets of the partially ordered set\(b\), the set of generic reals, can be systematically well ordered. He thenproves that in this model, the Cofinality Principle (51) and the OrderingPrinciple (30) are true, but the Order Extension Principle (49) is false.Forms 30 and 294 (Every linearly ordered \(W\)-set is well orderable.) areboth true in this model (51 implies 294). Since 30 \(+\) 294 \(+\) 144 (Everyset is a \(W\)-set.) implies AC, it follows that 144 is false. (See Note 25for definitions.) Moreover,Form 64 (There is no amorphous set.) is true(30 implies 64), soForm 133 (Every set is either well orderable or has aninfinite amorphous subset.) is false because 64 + 133 implies AC. TheBoolean Prime Ideal Theorem (14) is false because 14 implies 49, and\(C(\infty,<\aleph_0)\) (62) is true because 30 implies 62. It was shown byBlass (see Note 33) that 62 + 99 (Rado's Selection Lemma) implies 14, soform 99 is false.
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