Cohen \(\cal M27\): Pincus/Solovay Model I | Back to this models page
Description: Let \(\cal M_1\) be a model of \(ZFC + V =L\)
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. |
| 52 | Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\). |
| 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. |
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 |
|---|---|
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 206 | The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\). |
| 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. |
| 371 | There is an infinite, compact, Hausdorff, extremally disconnected topological space. Morillon [1993]. |
| 410 | RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology. |
Historical background: \(R\in 2^x\) is said to be random if \(R\in b\) for everyconstructible Borel set \(b\) which has Lebesgue measure 1. Let \(\cal M_2 =L(R)\), where \(R\) is a random element of \(2^{\omega_1^L}\). (Models likethis were also discussed in Sacks [1969] and Solovay [1971].) In \(\cal M_2\), for each \(\alpha\in \omega_1\), let\(r_\alpha=R|\alpha\cdot\omega\). Let \(G_\alpha^*\) be the set of all subsetsof \(\alpha\cdot\omega\) which are finite unions of half open intervals ofthe form \([\gamma\cdot \omega, (\gamma + 1)\cdot\omega)\), and \(G_\alpha\)is the set of subsets of \(\alpha\cdot\omega\) which differ finitely from amember of \(G_\alpha^*\). \(G_\alpha\) is an Abelian group under the operationof symmetric difference. Let \(F\) be the function on \(\omega_1\) such thatfor each \(\alpha\in\omega_1\), \(F(\alpha)\) is the orbit of \(r_\alpha\) underthe action of the group \(G_\alpha\). Then, \(\cal M27\) is the smallest modelof ZF containing \(F\). Solovay proves that the Principle of DependentChoices (43) and the Hahn-Banach Theorem (52) are true in \(\cal M27\), butall ultrafilters are principal in this model so 206 is false. 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.) Howard [1975] has shown that ifform 206 is false, then \L o\'s' Theorem (253) is true. Morillon [1993] has shown that ifForm 9 is true andForm 206 is false thenthere does not exist an infinite, compact, Hausdorff, extremallydisconnected topological space soForm 371 is false. It is shown inDelhomm\'e/Morillon [2000] thatForm 410 (The closed unit ballof a reflexive normed space is compact for the weak topology.) is false.
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