Cohen \(\cal M42\): Bull's Model | Back to this models page
Description: Let \(\cal M\) be a countable transitive model of \(ZFC +\) "There are uncountable regular cardinals \(\aleph_\alpha <\aleph_\beta < \aleph_\gamma\) such that \(\aleph_\alpha\) is \(\aleph_\gamma\)-supercompact; \(\aleph_\beta\) is the first measurable cardinal greater than \(\aleph_\alpha\); and \(\aleph_\gamma =|2^{\aleph_\beta}|\)." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of \(\cal M\)
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 |
|---|---|
| 39 | \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202. |
| 86-alpha | \(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) 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 |
|---|---|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 321 | There does not exist an ordinal \(\alpha\) such that \(\aleph_{\alpha}\) is weakly compact and \(\aleph_{\alpha+1}\) is measurable. |
Historical background: Then usingcollapsing functions, he constructs an inner model, \(\cal M42\), in which\(\aleph_\beta\) is collapsed to \(\aleph_{\alpha+1}\). He proves that\(\aleph_\alpha\) is an uncountable, weakly compact cardinal and\(\aleph_{\alpha+1}\) is measurable (321 is false). Bull also proves that\(C(\aleph_\alpha,\infty)\) (86) is true in this model. Since \(\alpha\ge 1\),form 39 (\(C(\aleph_1,\infty)\)) is also true. Form 163 (Every set iseither well orderable or has an infinite Dedekind finite subset.) is falsebecause 39 implies 8 and 8 + 163 implies AC (Brunner [1982a]).
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