Fraenkel \(\cal N6\): Levy's Model I | Back to this models page
Description: \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th 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. |
| 61 | \((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element 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). |
| 95-F | Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4. |
| 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 \(a\). |
| 218 | \((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\). |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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. |
| 171 | If \((P,\le)\) is a partial order such that \(P\) is the denumerable union of finite sets and all antichains in \(P\) are finite then for each denumerable family \({\cal D}\) of dense sets there is a \({\cal D}\) generic filter. |
| 308-p | If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup. |
| 314 | For every set \(X\) and every permutation \(\pi\) on \(X\) there are two reflections \(\rho\) and \(\sigma\) on \(X\) such that \(\pi =\rho\circ\sigma\) and for every \(Y\subseteq X\) if \(\pi[Y]=Y\) then \(\rho[Y]=Y\) and \(\sigma[Y]=Y\). (A reflection is a permutation \(\phi\) such that \(\phi^2\) is the identity.) \ac{Degen} \cite{1988}, \cite{2000}. |
| 334 | \(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of sets such that for all \(x\in X\), \(|x|\ge 2\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is even. |
| 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 \(E_i\). |
| 358 | \(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
| 379 | \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). |
Historical background: \(\cal G\) is thegroup generated by \(\{\pi_n : n\in\omega\}\), where, if \(P_n = \{a_{m+1},a_{m+2}, \cdots, a_{m+p_n}\}\), then\(\) \pi_n : a_{m+1}\mapsto a_{m+2}\mapsto\cdots a_{m+p_n}\mapsto a_{m+1}\hbox{ and } \pi_n(x)=x, \hbox{ for all }x\not\in P_n.\(\) \(S\) is theset of all finite subsets of \(A\). Levi proves in \(\cal N6\) that there isno choice function on \(\{P_n: n\in\omega\}\) so \(C(\aleph_0, <\aleph_0)\)(10) is false, but for each \(n\in\omega\), \(n>0\), \(C(\infty,n)\) (61) istrue. (It is also clear that \(\{P_n: n\in\omega\}\) has no infinite subsetwith a Kinna-Wagner selection function so \(KW(\aleph_0,<\aleph_0)\) (358)and \(PKW(\infty,\infty,\infty)\) are also false.) Levy, also proves theaxiom of Multiple Choice (67) is true. However, Bleicher has shown that\((\forall n\in\omega)MC(\infty, \infty,\) relatively prime to \(n\)) (218) isequivalent to 61 + 67, so 218 is also true. Since 218 implies 333(\(MC(\infty,\infty,\hbox{ odd})\)) and Keremedis has shown that \(333 + 334(MC(\infty,\infty,\hbox{ even})) \leftrightarrow\) AC, it follows that 334is false. Since 218 is true, and therefore, [218 A] (Existence ofComplementary Subspaces) is also true, it follows that 95(\(F\)) (Existenceof Complementary Subspaces over a Field \(F\)) is true.\parShannon proves if \(T =\{f: \exists n f\hbox{ is a choice function on }\{P_0,P_1,\cdots,P_n\}\}\) with the partial order \(f\le g\) iff \(g\subseteqf\), then \(T\) is a denumerable union of finite sets, all antichains arefinite, and there is a denumerable family of dense sets for which there inno generic filter (171 is false). See Note 47 for definitions. InHoward/Yorke [1987] it is shown that for any prime \(p\), there isa set of finite groups \(\{ G_y : y\in Y\}\) such that the weak directproduct has no maximal \(p\)-subgroup (308(\(p\)) is false in \(\cal N6\) forany prime \(p\)). SinceForm 106 (Baire category Theorem for compactHausdorff spaces) is true (218 implies 106) and 43 (Principle of DependentChoices) is false (43 implies 171),Form 154 (the Tychonoff theorem forcountably many \(T_2\) spaces.) must be false. (106 + 154 implies 43. SeeBrunner [1983c]) Degen has shown that if \(\rho_i = \pi_i/P_i\),\(\rho_i\) is not defined on \(A-P_i\), and \(\phi=\bigcup_{i\in\omega}\rho_i\), then \(\phi\) cannot be expressed as the product oftwo reflections. (\(\phi\) is a reflection if \(\phi^2=\phi\).) Consequently,form 314 (Every permutation on a non-empty set can be expressed as aproduct of two reflections.) is false.
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