Fraenkel \(\cal N50(E)\): Brunner's Model III | Back to this models page
Description: \(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)
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 |
|---|---|
| 6 | \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable. |
| 37 | Lebesgue measure is countably additive. |
| 67 | \(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 191 | \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). |
| 273 | There is a subset of \({\Bbb R}\) which is not Borel. |
| 305 | There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}. |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 322 | \(KW(WO,\infty)\), The Kinna-Wagner Selection Principle for a well ordered family of sets: For every well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15). |
| 361 | In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325. |
| 363 | There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325. |
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 |
|---|---|
| 10 | \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. |
| 154 | Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact. |
| 164 | Every non-well-orderable set has an infinite subset with a Dedekind finite power set. |
| 342-n | (For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\): Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.) |
| 344 | If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). |
Historical background: \(A\) is the disjoint union \(A = \bigcup_{p\in E,n\in\omega}A_{p,n}\). For each \(p\in E\) and \(n\in\omega\), let \(\eta_{p,n}\)be a permutation of \(A_{p,n}\) consisting of a single cycle of length\(p^n\). For each \(p\in E\) let \(\eta_p\) be the permutation of \(A\) definedby \(\eta_p(a) = \eta_{p,n}(a)\) if \(a\in A_{p,n}\) and \(\eta_p(a) = a\) for\(a\notin \bigcup_{n\in\omega} A_{p,n}\). \(\cal G\) is the group ofpermutations of \(A\) generated by \(\{\,\eta_p : p\in E\,\}\). \(S\) is theideal of finite subsets of \(A\). This is the model of the lemma on page 72of Brunner [1990]. In Brunner [1990] it is shown that\(MC(\infty,\infty)\) (form 67) and \(KW(WO,\infty)\) (form 322) are true andthat \(PC(\aleph_0,<\aleph_0, \infty)\) (form [10 E]) and\(PC(\infty,n,\infty)\) (form 45(\(n\))) (for \(n = \sum_{p\in E} n_p\cdot p\)where \(n_p\in \omega\) for \(p\in E\)) are false.
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