Fraenkel \(\cal N41\): Another variation of \(\cal N3\) | Back to this models page
Description: \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)
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. |
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 37 | Lebesgue measure is countably additive. |
| 63 |
\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 122 | \(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 144 | Every set is almost 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 |
|---|---|
| 88 | \(C(\infty ,2)\): Every family of pairs has a choice function. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 294 | Every linearly ordered \(W\)-set is well orderable. |
| 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 \(E_i\). |
| 350 | \(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). |
| 357 | \(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
Historical background: Thus, for each \(n\in\omega\), \((B_n,\le_n)\cong (\BbbQ, \le)\). \(\cal G\) is the group of all permutations on \(A\) such for all\(n\in\omega\), and all \(\phi\in\cal G\), \(\phi\) is an order automorphism of\((B_n,\le_n)\). \(S\) is the set of all finite unions of the \(B_n\)'s. Itfollows that \(A\) can be linearly ordered in this model, for if \(a\in B_m\)and \(b\in B_n\), \(a\le b\) if \(m < n\) or \(m=n\) and \(a\le_n b\). For each\(n\in\omega\), \(B_n\) can be well ordered so \(A\) is the union of a wellordered set of well orderable sets. Since \(A\) can be linearly ordered andnot well ordered, it follows that 294 is false. (See Note 25 fordefinitions.) Let \(C_j\) be the set of all pairs of countable subsets of\(B_j\), and let \(D=\bigcup_{j\in\omega}C_j\). Then \(D\) has no choicefunction. (Let \(\{E_j,O_j\}\in C_j\) such that \(E_j=\{a^j_{2i}: i\in\BbbZ\}\) and \(O_j=\{a^j_{2i+1}: i\in\Bbb Z\}\), where \(\Bbb Z\) is the set ofintegers. (If \(m, n\in\Bbb Q\), \(a^j_m\le a^j_n\) iff \(m\le n\).) If \(f\) werea choice function on \(D\) with support \(F\), then there is a\(\phi\in\hbox{fix}(F)\) and a \(j\in\omega\) such that \(\phi\) interchanges\(E_j\) and \(O_j\).) Thus, \(C(\infty,2)\) (88) is false. Also, in this modelevery infinite set has a countably infinite subset (9) is true and theaxiom of choice for well orderable families of finite sets (122) is true.See Note 112. Since 70 (There is a non-trivial ultrafilter on \(\omega\)) istrue in every FM model and 9 + 70 implies 63 (Every infinite set has anon-trivial ultrafilter.) it follows that 63 is true. Form 10(\(C(\aleph_0,<\aleph_0)\))is true (9 implies 10) andForm 231(\(UT(WO,WO,WO)\)) is false (231 implies 294). It follows thatForm 163(Every non well orderable set has an infinite Dedekind finite subset) isfalse. (10 + 163 implies 231. See Note 123.) The set \(\{B_n:n\in\omega\}\)is a denumerable set of denumerable sets that has no Kinna-Wagnerselection function and no multiple choice function so\(KW(\aleph_0,\aleph_0)\) (357) and \(MC(\aleph_0,\aleph_0)\) (350) are bothfalse. In Note 155 we show that ``every set is almost well orderable''(form 144) is true.\par
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