Fraenkel \(\cal N45(p)\): Howard/Rubin Model III | Back to this models page
Description: Let \(p\) be a prime
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. |
| 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 |
|---|---|
| 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(\(n\)). |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 220-p | Suppose \(p\in\omega\) and \(p\) is a prime. Any two elementary Abelian \(p\)-groups (all non-trivial elements have order \(p\)) of the same cardinality are isomorphic. |
Historical background: The set ofatoms \(A\) is a disjoint union,\(\)A = \bigcup_{i=1}^p B_i\cup\bigcup_{i=1}^p C_i\cup D\(\)where \(B_i=\{b_{n,\alpha}^i: n\in\omega\wedge\alpha <\aleph_1\}\),\(C_i=\{c_{n,\alpha}^i: n\in\omega\wedge\alpha <\aleph_1\}\), and\(D=\{d_{n,\alpha}:n\in\omega\wedge\alpha <\aleph_1\}\). For each \(\lambda<\aleph_1\), let\(\)A_{\lambda}= \bigcup_{i=1}^p B_{i,\lambda}\cup\bigcup_{i=1}^pC_{i,\lambda} \cup D_{\lambda}\(\) where\(B_{i,\lambda}=\{b_{n,\lambda}^i:n\in\omega\}\), \(C_{i,\lambda}=\{c_{n,\lambda}^i:n\in\omega\}\) and\(D_{\lambda}=\{d_{n,\lambda}:n\in\omega\}\). Let \(\phi_{\lambda}\) be thepermutation of \(A\) which is the product of the cycles,\(\)\phi_{\lambda}=\prod_{n\in\omega}(b_{n,\lambda}^0,b_{n,\lambda}^1,\cdots, b_{n,\lambda}^{p-1})(c_{n,\lambda}^0,c_{n,\lambda}^1, \cdots,c_{n,\lambda}^{p-1}).\(\) (Note that\(\phi_{\lambda}(d_{n,\lambda})=d_{n,\lambda}\) for each \(n\in\omega\) and\(\phi_{\lambda}(a)=a\), for each \(a\in A-A_{\lambda}\).) The group \(\cal G\)is the group generated by the \(\phi_{\lambda}\) for \(\lambda <\aleph_1\) and\(S\) is the set of all finite subsets of \(A\).An elementary Abelian \(p\)-group is an Abelian group such that everynon-identity element has order \(p\). (Note that an elementary Abelian\(p\)-group is just a vector space over the \(p\)-element field \(\{0,1,\cdots, p-1\}\).) Let \(G_1\) be the vector space over the \(p\)-elementfield \(\{0, 1,\cdots, p-1\}\) with basis \(\bigcup_{i=1}^p B_{i,\lambda}\)and \(G_2\) be the vector space over the same \(p\)-element field with basis\(\bigcup_{i=1}^p C_i\cup D\). It is shown that \(|G_1|=|G_2|\), but \(G_1\) and\(G_2\) are not isomorphic. Thus,Form 220(\(p\)) is false in this model.(This extends the result of Hickman [1977b] to \(p=2\) and 3. See\(\cal N42(p)\).)
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