Fraenkel \(\cal N23\): Howard/Mostowski Linearly Ordered Model | Back to this models page
Description: This model is amodification of \(\cal N3\)
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 |
|---|---|
| 8 | \(C(\aleph_{0},\infty)\): |
| 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 \(a\). |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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. |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 193 | \(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group. |
| 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: \(A\) is countably infinite; \(<\) is a denselinear ordering on \(A\) without first or last elements (\((A,<) \cong (\BbbQ,<)\)); \(\cal G\) is the group of all order automorphisms on \((A,<)\); and\(S\) is the set of all subsets of \(A\) that are well ordered by \(<\).P.~Howard has shown that AC for pairs (88) is false in this model andA.~Rubin has shown that AC for a denumerable number of sets (8) is true.See Note 100. It follows from Brunner [1982a] that in this modelthere is a set that cannot be well ordered and does not have an infiniteDedekind finite subset, (163 is false). (Form 8 plusForm 163 iff AC.)Since 8 implies 9 (Every Dedekind finite set is finite.) and 70 (There isa non-trivial ultrafilter on \(\omega\).) is true in every FM model, itfollows that 63 (Every infinite set has a non-trivial ultrafilter.) isalso true. Blass [1979] has shown that 191 (There is a set \(X\)such that for each set \(a\) there is an ordinal \(\alpha\) and a function \(f\)mapping \(X\times\alpha\) onto \(a\).) is true in every permutation model witha set of atoms. He has also shown that 8 + 191 + 193 (Every Abelian groupis a homomorphic image of a free projective Abelian group.) iff AC. Thus,193 is false. (See Note 60 for definitions.)
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