Fraenkel \(\cal N49\): De la Cruz/Di Prisco Model | Back to this models page
Description: Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)
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. |
| 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 \(a\). |
| 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. |
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 |
|---|---|
| 47-n | If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. |
| 106 | Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 167 | \(PKW(\aleph_{0},\ge 2,\infty)\), Partial Kinna-Wagner Principle: For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\). |
| 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 \(E_i\). |
| 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 \(y\). |
| 380 | \(PC(\infty,WO,\infty)\): For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function. |
Historical background: (\({\Bbb Q}/{\Bbb Z}\) denotes theadditive group.) For each \(i\in\omega\), let \(A_i = \{ a(i,p) : p\in {\BbbQ}/{\Bbb Z}\}\) and for each \(i\in\omega\) and \(p\in {\Bbb Q}/{\Bbb Z}\), let\(\pi^p_i : A \to A\) be defined by\(\pi^p_i(a(j,q)) =\cases a(j,p+q) &\hbox{if } j=i\\ a(j,q) &\hbox{otherwise}\endcases\).\(G\) is the group generated by the permutations \(\pi^p_i\) for \(i\in \omega\)and \(p\in {\Bbb Q}/{\Bbb Z}\). \(S\) is the set of finite supports. De laCruz and Di Prisco have shown that every infinite set in this model isDedekind infinite (9 is true) and that \(\{ A_i : i\in\omega \}\) is acountable family of well orderable sets such that no infinite subfamilyhas Kinna-Wagner selection function. It follows that 167, 379, and 380 arefalse. In addition De la Cruz and Di Prisco have shown that for every\(n\in\omega\), \(n\ge 2\), there is a set of \(n\) elements sets in the modelwith no choice function (47(n) is false).
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