Cohen \(\cal M3\): Mathias' model | Back to this models page
Description: Mathias proves that the \(FM\) model \(\cal N4\) can be transformed into a model of \(ZF\), \(\cal M3\)
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 |
|---|---|
| 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(\(n\)). |
| 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. |
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. |
| 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.) |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
Historical background: \(\cal M3\) is an extension of\(\cal M1\) in which a universal homogeneous partial ordering is defined on\(b\), the set of generic reals. (See Jech [1973b] p 101 fordefinitions.) Mathias then proves that the partial ordering on \(b\) cannotbe extended to a linear ordering (49 is false), but every set can belinearly ordered (30 is true). Also, in this model, every set of wellorderable sets has a choice function (60) and the Kinna-Wagner SelectionPrinciple (15) are 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 fordefinitions.) SinceForm 64 (There is no amorphous set.) is true (15implies 64),Form 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 15 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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