Cohen \(\cal M40(\kappa)\): Pincus' Model IV | Back to this models page
Description: The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
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 |
|---|---|
| 14 | BPI: Every Boolean algebra has a prime ideal. |
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 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. |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 87-alpha | \(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\). |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 295 | DO: Every infinite set has a dense linear ordering. |
| 345 | Rasiowa-Sikorski Axiom: If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). |
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 |
|---|---|
| 51 | Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 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. |
| 253 | \L o\'s' Theorem: If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. |
| 286 | Extended Krein-Milman Theorem: Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point. H. Rubin/J. Rubin [1985], p. 177-178. |
| 384 | Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter. |
Historical background: In \(\cal M\), \(\kappa\) is a regularcardinal larger than \(\aleph_0\) such that \(2^{\kappa}=\kappa^+\). Pincusconstructs a generic extension of \(\cal M\) using \(\kappa^+\) genericiterations. To assure that all the sets added in this manner havecanonical supports, he enumerates the sets at each level usinghereditarily almost disjoint sets. (See Pincus [1977a] p 125 fordefinitions.) The construction is made in such a way that at each level DC(43) holds, but AC does not. In Pincus [1977a] it is shown thatin this model, every set can be linearly ordered (30 is true), \(C(\infty,WO)\) (60) is true; and for all \(\aleph_{\alpha}<\kappa\), DC(\(\aleph_{\alpha}\)) (87(\(\alpha\))) is true. Pincus [1977b] proves thatthe Boolean Prime Ideal Theorem (14) is also true. Since 87 implies 8(\(C(\aleph_0,\infty)\)), it follows from Brunner [1982a] that inthis model there is a set that cannot be well ordered and does not have aninfinite Dedekind finite subset, (163 is false). (Form 8 plusForm 163 iffAC.)Form 14 implies 49 (Every partial ordering can be extended to alinear ordering.) and 49 + 51 (Every linear ordering has a cofinal subwell ordering.) iff AC. (See Morris [1969] and Note 121.) Thus,form 51 is false. SinceForm 30 is true andForm 87(\(\alpha\)) implies form296 (Every infinite set is the disjoint union of infinitely many infinitesets.), it follows from Pincus [1997] thatForm 295 (Everyinfinite set has a dense linear ordering.) is true. Since 43 + 14 isequivalent to 345 ([345 A]),Form 345 is true. Since 8 is true and 384 + 8implies 1,Form 384 is false.
Back