Fraenkel \(\cal N39\): Howard's Model II | Back to this models page
Description: \(A\) is denumerable and is a disjoint union\(\bigcup_{i\in\omega}B_i\cup\bigcup_{i\in\omega}C_i\), where for all\(i\in\omega, |B_i|=|C_i|=\aleph_0\)
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 |
|---|---|
| 16 | \(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function. |
| 23 | \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). |
| 24 | \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function. |
| 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. |
| 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 \(a\). |
| 231 | \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets 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 |
|---|---|
| 15 | \(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every 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 81(\(n\)). |
| 29 | If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup^{}_{x\in S} A_{x}| = |\bigcup^{}_{x\in S} B_{x}|\). Moore, G. [1982], p 324. |
| 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). |
| 131 | \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 328 | \(MC(WO,\infty)\): For every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.) |
Historical background: Let \(f_i\) be a fixed bijection from\(B_i\) onto \(C_i\) for each \(i\in\omega\). \(\cal G\) is the group of allpermutations \(\phi\) of \(A\) such that:\par\item{1.} \(\phi(B_i)=B_i\) and \(\phi(C_i)=C_i), \forall i\in\omega\).\item{2.} \(\{a\in A: \phi(a)\ne a\}\) is finite.\item{3.} \(\phi|B_i\) and \(\phi|C_i\) are even permutations,\(\forall i\in\omega\).\par\noindent For each finite \(x\subseteq\omega\) and each finite\(E\subseteq A\), define\(\)G(x,E)=\{\phi\in\hbox{ fix}(E): (\forall i\in x)(\phi(f_i)=f_i)\}\(\)\noindentLet \(\Gamma\) be the filter of subgroups of \(\cal G\) generated by the\(G(x,E)\), where \(x\) is a finite subset of \(\omega\) and \(E\) is a finitesubset of \(A\). (The filter is easier to describe for this model than thesupports.) \(\cal N39\) is the model determined by \(A\), \(\cal G\), and thefilter \(\Gamma\). It is shown in Howard [1994] that \(|B_i|=|C_i|\)for all \(i\in\omega\), but\(|\bigcup_{i\in\omega}B_i|\ne|\bigcup_{i\in\omega}C_i|\), so 29 is false.Howard also proves that the union of a well ordered collection of wellorderable sets is well orderable (\(UT(WO,WO,WO)\),Form 231) is true.However, in every FM model, 231 implies\((\forall\alpha)UT(\aleph_{\alpha},\aleph_{\alpha},\aleph_{\alpha})\) soform 23 is also true. \(C(WO,<\aleph_0)\) (122) is true (231 implies 122)and 122 + 67 (MC) implies AC (see [67 B]), so it follows that MC is false.SinceForm 165 (\(C(WO,WO)\)) is true, (231 implies 165) it follows fromNote 2(8 and 9) that 16 and 24 are also true.
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