Cohen \(\cal M2(\langle\omega_2\rangle)\): Feferman/Truss Model | Back to this models page
Description: This is another extension of \(\cal M2\)
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 |
|---|---|
| 181 | \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) 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 |
|---|---|
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 202 | \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function. |
| 203 | \(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function. |
Historical background: The set of all sequences of generic reals oflength less than \(\omega_2\) are added to a countable transitive model ofZFC + V = L. Truss proves that \(C(|\Bbb R|,\infty)\) (181) is true, butthere is a partition of \(\cal P(\omega)\) that has no choice function (203is false). Since the Axiom of choice is false, it follows that all formsthat imply AC in ZF are also false. In particular, \(C(LO,\infty)\) (202),is false. (See the first paragraph of section I.)Form 163 (every set iseither well orderable or has an infinite Dedekind finite subset) is alsofalse because 181 implies 8 and 8 + 163 implies AC (Brunner [1982a]).
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