Cohen \(\cal M29\): Pincus' Model II | Back to this models page
Description: Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)
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 |
|---|---|
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 200 | For all infinite \(x\), \(|2^{x}| = |x!|\). |
| 214 | \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). |
| 290 | For all infinite \(x\), \(|2^x|=|x^x|\). |
| 291 | For all infinite \(x\), \(|x!|=|x^x|\). |
| 295 | DO: Every infinite set has a dense linear ordering. |
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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 32 | \(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 118 | Every linearly orderable topological space is normal. Birkhoff [1967], p 241. |
| 151 | \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).) |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 350 | \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). |
| 357 | \(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
Historical background: In this model\(\{I_n: n\in\omega\}\) is a denumerable set of denumerable sets which hasneither a choice function (32 is false), nor a Kinna-Wagner selectionfunction (\(KW(\aleph_0,\aleph_0)\), 357 is false), nor a multiple choicefunction (\(MC(\aleph_0,\aleph_0)\), 350 is false); \(|I| < |2\times I|\) (3is false); every set can be linearly ordered (30 is true); every infiniteset has a denumerable subset (9 is true); and for all infinite \(x\), \(|x!|= |2^x| = |x^x|\) (200, 290, and 291 are true). Pincus also proves thatif \(x\) is a set of infinite sets, then there is a function \(f\) with domain\(x\) such that for each \(y\in x\), \(f(y)\subseteq y\), and \(|f(y)|=\aleph_0\)(214 is true). In this modelForm 10 (\(C(\aleph_0,<\aleph_0)\)) is true (30implies 10) andForm 165 (\(C(WO,WO)\)) is false (165 implies 32). Thus,form 163 (Every non well orderable set has an infinite Dedekind finitesubset.) is false because Brunner [1982a] shows that 10 + 163implies 165. SinceForm 30 is true andForm 214 impliesForm 296 (Everyinfinite set is the disjoint union of infinitely many infinite sets.), itfollows from Pincus [1997] thatForm 295 (Every infinite set hasa dense linear ordering.) is true.
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