Cohen \(\cal M25\): Freyd's Model | Back to this models page
Description: Using topos-theoretic methods due to Fourman, Freyd constructs a Boolean-valued model of \(ZF\) in which every well ordered family of sets has a choice function (Form 40 is true), but \(C(|\Bbb R|,\infty)\) (Form 181) 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 |
|---|---|
| 40 | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets 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 |
|---|---|
| 67 | \(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 181 | \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function. |
Historical background: Freyd describes an algebraic theory with twoconstants, three unary operations, and two binary operations. (SeeFreyd [1980], p 112.) Let \(F\) be the free algebra generated bythe two constants and let \(\cal B(F)\) be the corresponding category ofBoolean sheaves. The Fourman interpretation arising from \(\cal B(F)\) isthe desired model. (See, for example, MacLane [1971] andJohnstone [1977] for background information on category theory.)Since 40 implies 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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