Cohen \(\cal M11\): Forti/Honsell Model | Back to this models page
Description: Using a model of \(ZF + V = L\) for the ground model, the authors construct a generic extension, \(\cal M\), using Easton forcing which adds \(\kappa\) generic subsets to each regular cardinal \(\kappa\)
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 |
|---|---|
| 15 | \(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)). |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 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 |
|---|---|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 105 | There is a partially ordered set \((A,\le)\) such that for no set \(B\) is \((B,\le)\) (the ordering on \(B\) is the usual injective cardinal ordering) isomorphic to \((A,\le)\). |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 144 | Every set is almost well orderable. |
| 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. |
Historical background: (See Easton [1970].) Then, \(\cal M11\) is asymmetric submodel of \(\cal M\). In this model, the Kinna-Wagner Principle(15) and the Axiom of Choice for a family of well orderable sets (60) aretrue, but for any partially ordered set \((X, \precsim)\) there is abijection \(f : X \longrightarrow Y\) such that for all \(x, y \in X\),\(x\precsim y \leftrightarrow |f(x)| \le |f(y)|\) (105 is false). Since form64 (There is no amorphous set.) is true (30 implies 64),Form 133 (Everyset is either well orderable or has an infinite amorphous subset.) isfalse. Form 163 (Every set is either well orderable or has an infiniteDedekind finite subset.) is also false because 43 implies 8 and 8 + 163implies AC (Brunner [1982a]).Form 15 implies the OrderingPrinciple (30) andForm 60 implies that every linearly ordered set that isalmost well orderable is well orderable (294). (See Note 25 fordefinitions.) However, 30 + 144 (Every set is almost well orderable.) +231 iff AC. Thus,Form 144 is false.
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