Cohen \(\cal M18\): Shelah's Model I | Back to this models page
Description: Shelah modified Solovay's model, \(\cal M5\), and constructed a model without using an inaccessible cardinal in which the Principle of Dependent Choices (Form 43) is true and every set of reals has the property of Baire (Form142 is false)
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. |
| 93 | There is a non-measurable subset of \({\Bbb R}\). |
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 |
|---|---|
| 142 | \(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
Historical background: Stern has shown that in this modelthere is a non-measurable set of real numbers (93 is true). Since 43implies 8 (\(C(\aleph_0,\infty)\)), it follows from Brunner [1982a] that in this model there is a set that cannot be well orderedand does not have an infinite Dedekind finite subset, (163 is false).(Form 8 plusForm 163 iff AC.)
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