Cohen \(\cal M38\): Shelah's Model II | Back to this models page
Description: In a model of \(ZFC +\) "\(\kappa\) is a strongly inaccessible cardinal", Shelah uses Levy's method of collapsing cardinals to collapse \(\kappa\) to \(\aleph_1\) similarly to Solovay [1970]
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 |
|---|---|
| 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. |
| 142 | \(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7. |
| 169 | There is an uncountable subset of \({\Bbb R}\) without a perfect subset. |
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 |
|---|---|
| 93 | There is a non-measurable subset of \({\Bbb R}\). |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
Historical background: (See \(\cal M5\) and \(\cal M12(\aleph)\).) He then builds aspecial forcing notion and constructs a model in which the Principle ofDependent Choices (43) is true and every set of reals is Lebesguemeasurable (93 is false) just as in \(\cal M5\). However, unlike \(\cal M5\),in Shelah's model there is a set of reals without the Baire property (142is true) and there is an uncountable set of reals with no perfect set (169is true). Since 43 implies 8 (\(C(\aleph_0,\infty)\)), it follows fromBrunner [1982a] that in this model there is a set that cannot bewell ordered and does not have an infinite Dedekind finite subset, (163 isfalse). (Form 8 plusForm 163 iff AC.)
Back