Fraenkel \(\cal N21(\aleph_{\alpha+1})\): Jensen's Model | Back to this models page
Description: We assume \(\aleph_{\alpha+1}\) is a regular cardinal
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 |
|---|---|
| 39 | \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202. |
| 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. |
| 86-alpha | \(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function. |
| 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 |
|---|---|
| 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(\(n\)). |
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 106 | Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire. |
| 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. |
| 193 | \(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group. |
Historical background: Let \(A=\{(a_{i_1}, a_{i_2},\dots, a_{i_n}) : n\in\omega\wedge(\forall j\le n) (i_j<\aleph_{\alpha+1})\}\). \(A\) is partially ordered by extension. Let \(\cal G\)be the group of all permutations of \(A\) that preserves the partialordering, and let \(S\) be the set of all subsets \(E\) of \(A\) that satisfythe following properties:\item{(1)} The cardinality of \(E\) is less than \(\aleph_{\alpha+1}\);\item{(2)} If \(s\subseteq t\) and \(t\in E\), then \(s\in E\); and\item{(3)} There is no function \(f : \omega\to \aleph_{\alpha+1}\)\(f|n\in E\) for all \(n\in\omega\). \par\noindentIn this model, the Principle of Dependent Choices (43) is false. The Axiomof choice for a family of \(\aleph_{\beta}\) sets (86(\(\beta\))) is true forall \(\beta < \alpha+1\) and, therefore, the Axiom of Choice for a familyof \(\aleph_1\) sets (39) is true for all \(\alpha \ge 1\). For all cardinals\(\kappa\), \(\kappa\) is comparable with \(\aleph_{\beta}\) (71(\(\beta\))) istrue for each \(\beta < \alpha+1\). Since 39 implies 9 (Every Dedekindfinite set is finite.) and 70 (There is a non-trivial ultrafilter on\(\omega\).) is true in every FM model, it follows that 63 (Every infiniteset has a non-trivial ultrafilter.) is also true. Since 39 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.) Blass [1979] has shown that 191 (There is a set \(X\) suchthat 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. Since8 and 191 are true, 193 must be false. (See Note 60 for definitions.)
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