Fraenkel \(\cal N28\): Blass' Permutation Model | Back to this models page
Description: The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\)
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. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 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}. |
| 309 | The Banach-Tarski Paradox: There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\). |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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. |
| 368 | The set of all denumerable subsets of \(\Bbb R\) has power \(2^{\aleph_0}\). |
| 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 |
|---|---|
| 129 | For every infinite set \(A\), \(A\) admits a partition into sets of order type \(\omega^{*} + \omega\). (For every infinite \(A\), there is a set \(\{\langle C_j,<_j \rangle: j\in J\}\) such that \(\{C_j: j\in J\}\) is a partition of \(A\) and for each \(j\in J\), \(<_j\) is an ordering of \(C_j\) of type \(\omega^* + \omega\).) |
| 190 | There is a non-trivial injective Abelian group. |
| 293 | For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. |
Historical background: For each \(\xi\in\aleph_1\), let \(g_{\xi}\) be the permutation of \(A\) such that \(\)g_{\xi}(a^i_{\eta}) = \cases a^{i+1}_\eta&\hbox{if \(\eta =\xi\)}\\a^i_{\eta} &\hbox{if \(\eta\not=\xi\)}\endcases\(\)\noindentLet \(\cal G\) be the group of permutations generated by all the \(g_{\xi}\)'sfor \(\xi\in\aleph_1\). Let \(\)S =\{\{a^i_{\xi}: i\in\Bbb Z, \xi\in E\} : E\hbox{ a finite subset of }\aleph_1\}.\(\) Let \(\cal N\) be the permutation model generated by \(A\), \(\cal G\), and \(S\). For any set \(x\), \(x\) is calledsmall if \(A\cap TC(x)\) is countable. (\(TC(x)\) is the transitive closure of \(x\).) Let the universe of \(\cal N28\) be the class of all small sets in \(\cal N\). Blass shows that \(\cal N28\) is a model of ZF\(^0\) in which \(A\) is a proper class and he also shows that there is no non-trivial injective Abelian group in this model (190 is false). Blass also proves thatForm 190 is true in every permutation model in which the class of atoms is a set. Let \(A' = \left\{ a_\xi^i : i \in {\Bbb Z} \land \xi \in\aleph_0 \,\right\}\). \(A'\) is a denumerable subset of the atoms (in theground model) and is therefore a set in the model. Further, \(A'\) islinearly orderable in \(\cal N28\). For each \(\xi\in \aleph_0\), let \(O_\xi =\{ a^{2i}_\xi : i \in {\Bbb Z} \,\}\) and \(E_\xi = \{ a^{2i+1}_\xi : i \in{\Bbb Z} \,\}\). The set \(B = \left\{ \,O_\xi : \xi\in\aleph_0 \,\right\}\cup \left\{\, E_\xi : \xi\in\aleph_0 \, \right\}\) cannot be linearlyordered in \(\cal N28\) soForm 293 is false.Form 129 is false because theset \(B\) cannot be partitioned into sets of order type \(^*\omega + \omega\).(See model \(\cal N37\).)
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