Cohen \(\cal M10\): Derrick/Drake Model | Back to this models page
Description: Let \(\cal M\) be a model of \(ZF + GCH\). Add to \(\cal M\) generic functions \(f_n\) for each \(n\in\omega\), where \(f_n:\omega_n\to\cal P(\omega)\), but do not add \(\{f_n: n\in\omega\}\)
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 |
|---|---|
| 0 | \(0 = 0\). |
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 |
|---|---|
| 40 | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 56 |
\(\aleph(2^{\aleph_{0}})\neq\aleph_{\omega}\). (\(\aleph(2^{\aleph_{0}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph_{0}}\).) |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 275 | The sequence of cardinals \(\langle\aleph_n: n \in\omega\rangle\) has a unique minimal upper bound. |
Historical background: It isshown that in this model \(\aleph(2^{\aleph_0})= \aleph_\omega\) (56 isfalse). (\(\aleph(x)\) is the smallest ordinal that is not similar to asubset of \(x\), Hartogs' function.) (In fact, for every \(\alpha>0\), it isconsistent with ZF that \(\aleph(2^{\aleph_0}) = \aleph_\alpha\).) It isshown by Truss that in \(\cal M10\) the sequence \(\{\aleph_n : n\in\omega\}\)has two distinct minimal upper bounds \(2^{\aleph_0}\) and\(\aleph_{\omega}\). (275 is false). Truss also proves that there is a wellordered set of sets that does not have a choice function (40 is false) andthat the Principle of Dependent Choices (43) is false.
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