Fraenkel \(\cal N54\): Good/Tree/Watson Model II | Back to this models page
Description: This model is a variation of \(\cal N 53\)
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. |
| 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. |
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(\(n\)). |
| 173 | \(MPL\): Metric spaces are para-Lindelöf. |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
Historical background: The set of atoms \(A=\bigcup \{Q_n:\ n\in \omega\}\), where\(Q_n=\{a_{n,q}:q\in \Bbb{R}\}\). We let \(<\) be the lexicographic orderingon \(A\). The group of permutations \(\cal G\), is the group of allpermutations on \(A\) which are a translation on \(Q_n\), that is, if \(\phi\in \cal G\), then \(\phi|Q_n(a_{n,q})=a_{n,q+r_n}\) for some \(r_n\in\Bbb R\).Supports are finite. Let \(d\) be the metric on \(A\) given by:\(\)d(a_{n,q},a_{m,p})= 1\hbox{ if } n\ne m\(\) and\(\)d(a_{n,q},a_{m,p})= |q-p|/(1+|q-p|) \hbox{ if }n=m.\(\)It is shown in Howard/Keremedis/Rubin/Stanley [1999] that \((A,d)\) is a metric space that is not meta-Lindel\"of (173 is false).
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