Fraenkel \(\cal N12(\aleph_{\alpha})\): A generalization of \(\cal N12(\aleph_1)\).Replace ``\(\aleph_1\)'' by ``\(\aleph_{\alpha}\)'' where \(\aleph_{\alpha}\) isa singular cardinal | Back to this models page
Description: Thus, \(|A|=\aleph_{\alpha}\); \(\cal G\) is the groupof all permutations on \(A\); and \(S\) is the set of all subsets of \(A\) withcardinality less than \(\aleph_{\alpha}\)
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 |
|---|---|
| 63 |
\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
|
| 71-alpha | \(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119. |
| 87-alpha | \(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\). |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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 \(m\), \(2m = m\). |
| 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(\(n\)). |
| 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. |
Historical background: Jech shows that\(DC(\aleph_{\beta})\) holds for all \(\aleph_{\beta}<\)cf(\(\aleph_ {\alpha}\))(87(\(\beta\)) for \(\aleph_\beta<\hbox{ cf}(\aleph_\alpha)\) is true);\(W_{\aleph_{\beta}}\) (For all cardinals \(m\), \(m\le\aleph_\beta\) or\(\aleph_\beta\le m\).) holds for all \(\beta<\alpha\) (71(\(\beta\)) is truefor \(\beta<\alpha\)); but 71(\(\alpha\)) and \(C(\infty,\hbox{ cf}\aleph_\alpha)\) are false. It follows from Note 104 that the \(2m=m\)principle (3) is false. Since 87 implies 8 (\(C(\aleph_0,\infty)\)), itfollows from Brunner [1982a] that in this model there is a setthat cannot be well ordered and does not have an infinite Dedekind finitesubset, which implies that 163 is false. (Form 8 plusForm 163 iff AC.)Since 87(\(\beta\)) 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. Thus,Form 193 must be false if \(\aleph_{\beta}<\)cf\((\aleph{\alpha})\). (See Note 60 for definitions.)
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