Cohen \(\cal M30\): Pincus/Solovay Model II | Back to this models page
Description: In this construction, an \(\omega_1\) sequence of generic reals is added to a model of \(ZFC\) in such a way that the Principle of Dependent Choices (Form 43) is true, but no nonprincipal measure exists (Form 223 is false)
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. |
| 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. |
| 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 \(\forall A\subseteq G, \forall g \in G\), \(\mu(gA)=\mu(A)\).) |
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. |
| 223 | There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\). |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
Historical background: (The construction is similar to that givenfor \(\cal M27\).) Since 43 implies 8 (\(C(\aleph_0,\infty)\)), it followsfrom Brunner [1982a] that in this model there is a set thatcannot be well ordered and does not have an infinite Dedekind finitesubset (163 is false). (Form 8 plusForm 163 iff AC.) Howard [1975] has shown that ifForm 206 (There exists a nonprincipalultrafilter.) is false (206 implies 223), then \L os' Theorem (253) istrue. By results of Wagon [1985], the amenable groups in \(\calM30\) are the finite groups and therefore 312 (A subgroup of an amenablegroup is amenable.) is true and 313 (\(\Bbb Z\) (the set of integers underaddition) is amenable.) is false. (\(G\) is amenable if there is a finitelyadditive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forallA\subseteq G, \forall g \in G\), \(\mu(gA)=\mu(A)\).)
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