Fraenkel \(\cal N56\): Howard's model III: Assume the the atoms are indexed asfollows: \(A = \{a(i,j) : i\in{\Bbb Q} \hbox{ and } j\in\omega \}\) | Back to this models page
Description: Foreach \(i\in \Bbb Q\), let \(A_i = \{a(i,j) : j\in \omega\}\)
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. |
| 40 | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 92 | \(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 182 | There is an aleph whose cofinality is greater than \(\aleph_{0}\). |
| 189 | \(EI\ Ab\): For every Abelian group \(A\) there is an injective Abelian group \(G\) and a one to one homomorphism from \(A\) into \(G\). |
| 190 | There is a non-trivial injective Abelian group. |
| 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}. |
| 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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 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(\(n\)). |
| 62 | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
| 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). |
| 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.) |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
Historical background: (call \(A_i\) the\(i\)th block.) Let\(\)\aligned \cal G = \{ \phi:A\to A : &(\forall i\in\Bbb Q)(\exists i'\in\Bbb Q)(\phi(A_i) = A_{i'})\hbox{ and }\\ &\{i\in\Bbb Q : (\exists j)(\phi(a(i,j))\ne a(i,j)) \}\hbox{ is bounded}\}.\endaligned\(\)\noindent Here bounded means bounded in the usual ordering on \(\Bbb Q\).(\(\cal G\) is the set of all permutations of \(\phi\) of \(A\) such that \(\phi\)of any block is a block and the set of blocks on which \(\phi\) is not theidentity is bounded.) \(S\) is the set of subsets of \(A\) of the form\(\bigcup_{i\in E} A_i\) where E is a bounded subset of \(\Bbb Q\). Form 40(\(C(WO,\infty)\)) is true using a proof similar to the proof of 40 in \(\calN33\) given in Howard/Rubin/Rubin [1973]. By Note 104,Form 3(\(2m=m\)) is false. Form 62 (\(C(\infty,<\aleph_0)\)) is false since thefamily of finite, non-empty subsets of \(A\) has no choice function.Form 67(MC) is false since the set of blocks has no multiple choice function.
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