Fraenkel \(\cal N35\): Truss' Model IV | Back to this models page
Description: The set of atoms, \(A\), is denumerable andeach element of \(A\) is associated with a finite sequence of zeros andones
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 \(a\). |
| 250 | \((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function. |
| 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 |
|---|---|
| 249 | If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch. |
| 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 \(E_i\). |
Historical background: We define a partial ordering \(\precsim\) on \(A\) such that if \(a_i,a_j\in A\), \(a_i\precsim a_j\) if \(a_i\) is an initial segment of \(a_j\).Then, \((A,\precsim)\) is a tree. \(\cal G\) is the group of permutation of\(A\) which is generated by the set \(\{\sigma_a: a\in A\}\) where if \(\tau\)is the transposition \((0,1)\) and \(a\precsim b\) then if \(b=\langlea,i,b'\rangle\), then \(\sigma_a(b)=\langle a,\tau(i),b'\rangle\), otherwise,\(\sigma_a(b)=b\). For this model, it is easier to describe the filter \(F\)of subgroups of \(G\) rather than the ideal of supports \(S\). \(F\) isgenerated by \(\{\cal G_m: m\in\omega\}\) where \(\cal G_0=\cal G\), and \(\calG_{m+1}=\bigcap\{K: K\hbox{ is a subgroup of }\cal G_m\wedge |\cal G_m:K|=2\}\). In this model \((A,\precsim)\) is a binary tree with no infinitebranch (249) is false, but for each \(n\in\omega\) the Axiom of Choice for awell ordered family of \(n\)-element sets (250) is true.
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