Fraenkel \(\cal N33\): Howard/H\.Rubin/J\.Rubin Model | Back to this models page
Description: \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all boundedsubsets of \(A\)
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 |
|---|---|
| 23 | \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). |
| 40 | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 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. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 62 | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
| 133 | Every set is either well orderable or has an infinite amorphous subset. |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 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. |
| 260 | \(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element. |
| 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 \(E_i\). |
| 349 | \(MC(\infty,\aleph_0)\): For every 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 \(x\). |
| 356 | \(KW(\infty,\aleph_0)\), The Kinna-Wagner Selection Principle for a family of denumerable sets: For every 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: It is shown that the axiom of choice for a well orderedset of non-empty sets (40) is true. (Since 40 implies \(UT(WO,WO,WO)\)(231) and in every FM model 231 implies\((\forall\alpha)UT(\aleph_{\alpha}, \aleph_{\alpha}, \aleph_{\alpha})\)(23),Form 23 is true.) However, a set \(x\subseteq A\times A\) and atransitive relation \(R\) on \(x\) are constructed so that every subset of\(x\) on which \(R\) is antisymmetric has an upper bound, but \(x\) has nomaximal element (260 is false). AC for a set of well orderable sets (60)is false. (In every FM model, 40 + 60 \(\to\) AC, Howard/Rubin [1995b].) In every FM model \(C(\infty,<\aleph_0)\) (62) implies 60(Howard [1973]). Thus, 62 is false. Since 40 implies 8(\(C(\aleph_0,\infty)\)), it follows from Brunner [1982a] that inthis model there is a set that cannot be well ordered and does not have aninfinite Dedekind finite subset, (163 is false). (Form 8 plusForm 163 iffAC.) Since 8 implies 9 (Every Dedekind finite set is finite.) and 70(There is a non-trivial ultrafilter on \(\omega\).) is true in every FMmodel, it follows that 63 (Every infinite set has a non-trivialultrafilter.) is also 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 inevery permutation model with a set of atoms. He has also shown that 8 +191 + 193 (Every Abelian group is a homomorphic image of a free projectiveAbelian group.) iff AC. Since 40 implies 8,Form 193 must be false. (SeeNote 60 for definitions.) Form 3 (\(2m=m\)) is false because \(2|A|\ne |A|\).Forms 349 (\(MC(\infty,\aleph_0)\)) and 356 (\(KW(\infty, \aleph_0)\)) arealso false because the set of all infinite bounded intervals of \(A\) is aset of denumerable sets which has no multiple choice function and noKinna-Wagner selection function.
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