Fraenkel \(\cal N36(\beta)\): Brunner/Howard Model III | Back to this models page
Description: This model is a modificationof \(\cal N15\)
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. |
| 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. |
| 63 |
\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
|
| 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. |
| 122 | \(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 144 | Every set is almost 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(\(n\)). |
| 67 | \(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). |
| 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. |
Historical background: \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_{\beta+1}\}\). Let \(\Gamma\) be the group of allpermutations on \(\omega\). Then \(\cal G=\{g:(\forall\alpha\in\omega_{\beta+1}) (\exists\gamma\in\Gamma)(\foralli\in\omega) g(a_{i,\alpha})=a_{\gamma(i),\alpha}\) and for all but finitelymany \(\alpha\in\omega_{\beta+1},(\forall i\in\omega) g(a_{i,\alpha}) =a_{i,\alpha}\}\). \(S\) is the set of all subsets of \(A\) of cardinality atmost \(\aleph_{\beta}\). This is the model of proposition 3.3 inBrunner/Howard [1992]. In this model \(C(\aleph_{\beta},\infty)\)(86(\(\beta\)) is true and, therefore, \(C(\aleph_1,\infty)\) (39) is true if\(\beta\ge 1\). The union of a denumerable set of finite sets is wellorderable ([122 A] is true), but there is an infinite set that is not theunion of a disjoint well orderable family of denumerable sets (152 isfalse). Since 39 implies 8 (\(C(\aleph_0,\infty)\)), it follows fromBrunner [1982a] that in this model there is a set that cannot bewell ordered and does not have an infinite Dedekind finite subset, (163 isfalse). (Form 8 plusForm 163 iff AC.) Since 8 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. Blass [1979]has shown that 191 (There is a set \(X\) such that for each set \(a\) there isan ordinal \(\alpha\) and a function \(f\) mapping \(X\times\alpha\) onto \(a\).)is true in every permutation model with a set of atoms. He has also shownthat 8 + 191 + 193 (Every Abelian group is a homomorphic image of a freeprojective Abelian group.) iff AC. Thus,Form 193 must be false. (SeeNote 60 for definitions.) In Note 144 we show dependent choice (form 43)is true and in Note 155 we show that ``every set is almost wellorderable'' (form 144) is true.
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