Cohen \(\cal M5(\aleph)\): Solovay's Model | Back to this models page
Description: An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)
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. |
| 307 | If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\). |
| 369 | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
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 |
|---|---|
| 88 | \(C(\infty ,2)\): Every family of pairs has a choice function. |
| 93 | There is a non-measurable subset of \({\Bbb R}\). |
| 142 | \(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 169 | There is an uncountable subset of \({\Bbb R}\) without a perfect subset. |
| 224 | There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\),
|
| 234 | There is a non-Ramsey set: There is a set \(A\) of infinite subsets of \(\omega\) such that for every infinite subset \(N\) of \(\omega\), \(N\) has a subset which is in \(A\) and a subset which is not in \(A\). |
| 277 | \(E(D,VII)\): Every non-well-orderable cardinal is decomposable. |
| 280 | There is a complete separable metric space with a subset which does not have the Baire property. |
| 281 | There is a Hilbert space \(H\) and an unbounded linear operator on \(H\). |
| 312 | A subgroup of an amenable group is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g \in G\), \(\mu(gA)=\mu(A)\).) |
| 313 | \(\Bbb Z\) (the set of integers under addition) is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 318 | \(\aleph_1\) is not measurable. |
Historical background: (The method ofcollapsing cardinals that is used in the model and some of the proofs aredue to A.~Levy. An inaccessible cardinal is used to construct the model.See, for example, Drake [1974], Kanamori/Magador [1978] or Solovay/Reinhardt/Kanamori [1978] for informationabout large cardinals.) In this model, The Principle of Dependent Choices(43) is true and every set of reals is Lebesgue measurable (93 is false).Moreover, if \(\aleph\) is a measurable cardinal, then so is \(\aleph_1\) (318is false). (Also, see Jech [1968a].) (It has been shown byShelah [1984] and Raisonnier [1984] that if ZF +\(C(\aleph_0,\infty)\) (8) + \(\neg\) 93 is consistent (43 implies 8), thenso is ZFC + ``there exists an inaccessible cardinal''. Thus, the existenceof an inaccessible cardinal is necessary for this result.) Moreover, in\(\cal M5(\aleph)\), every set of reals has the Baire property (142 isfalse) and the cardinal \(2^{\aleph_0}\) is not well orderable and is notdecomposable (the sum of two smaller positive cardinals) so 277 is falseand 369 is true. Also, the Hahn-Banach Theorem (52) and the Axiom ofChoice for pairs (88) are false. Every uncountable set of reals containsa perfect subset (169 is false). The following two statements are alsofalse in this model: There is a complete separable metric space with anon meager subset (280) and there is a Hilbert space \(H\) and an unboundedlinear operator on \(H\) (281). (See also Note 96.) Stern proves that inthis model, every infinite sequence of Borel sets of the same rank iscountable (224 is false). (Truss has shown that if the same constructionis used with \(\aleph\) a limit cardinal then 169 is still false, but 43 isalso false. See \(\cal M12(\aleph)\).) Mathias has shown that there is nonon-Ramsey set (234 is false). According to Kanovei [1991], if\(m\) is the cardinality of the set of Vitali equivalence classes, then, in\(\cal M5(\aleph)\), \(H(m)= H(2^{\aleph_0})\), where \(H\) is Hartogs' function(307 is true). (\(H(x)\) is the smallest ordinal that is not similar to asubset of \(x\) and the Vitali equivalence classes are equivalence classesof the real numbers under the relation \(x\equiv y \leftrightarrow (\existsq\in {\Bbb Q})(x-y=q)\).) Since 43 implies 8 (\(C(\aleph_0,\infty)\)), itfollows from Brunner [1982a] that in this model there is a setthat cannot be well ordered and does not have an infinite Dedekind finitesubset, (163 is false). (Form 8 plusForm 163 iff AC.) In Wagon [1985], it is shown that \(\Bbb Z\) is not amenable in \(\calM5(\aleph)\) (313 is false) and that there is an amenable group with anon-amenable subgroup in \(\cal M5(\aleph)\) (312 is false). (\(G\) isamenable if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) suchthat \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\),\(\mu(gA)=\mu(A)\).)
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