Fraenkel \(\cal N34\): Pincus' Model X | Back to this models page
Description: For each \(q\in\Bbb Q\), let \(C_q=\{a_q,b_q\}\), a pair of atoms and let \(A=\bigcup_{q\in\Bbb Q}C_q\)
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 \(a\). |
| 216 | Every infinite tree has either an infinite chain or an infinite antichain. |
| 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 |
|---|---|
| 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(\(n\)). |
| 217 | Every infinite partially ordered set has either an infinite chain or an infinite antichain. |
Historical background: Let \(\eusb C= \{ C_q \, : q \in {\Bbb Q} \,\}\) and let \(\preceq\) be the ordering\(\eusb C\) inherits from \(\Bbb Q\). (I. e., \(C_q \preceq C_r\Leftrightarrow q \le r\) where \(\le\) is the usual ordering on \(\Bbb Q\).)Let \(\cal G\) be the group of permutations of \(A\) generated by the the twogroups\(\)\alignG_1 = \{ \phi :& (\exists f)[ f \hbox{ is an order automorphism of } {\BbbQ}\\& \land (\forall q \in {\Bbb Q})( \phi(a_q) = a_{f(q)} \land \phi(b_q) = b_{f(q)})]\,\}\endalign\(\)and\(\)\alignG_2 = \{ \psi :& \psi \hbox{ is a finite product of transposition, }\\& \psi = (a_{q_1},b_{q_1})\circ \cdots \circ (a_{q_n},b_{a_n}) \hbox{ forsome } q_1, \ldots, q_n \in {\Bbb Q}\,\}.\endalign\(\)\(G_1\) could also be described as \(\{ \phi\, : \phi : A \to A\) is one toone and onto and (the extension of) \(\phi\) is an order isomorphism of\((\eusb C, \preceq)\,\}\). \(S\) is the set of finite subsets of \(A\). Inthis model, each infinite tree has either an infinite chain or an infiniteantichain (216 is true), but there is an infinite partially ordered setthat has neither an infinite chain nor an infinite antichain (217) isfalse. (See Note 105.)
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