Fraenkel \(\cal N12(\aleph_1)\): A variation of Fraenkel's model, \(\cal N1\) | Back to this models page
Description: Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false
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 |
|---|---|
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 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.
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| 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. |
| 187 | Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.) |
| 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 \(a\). |
| 202 | \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function. |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
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 \(m\), \(2m = m\). |
| 41 | \(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \). |
| 45-n | If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. |
| 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). |
| 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. |
| 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. |
| 323 | \(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.) |
| 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 \(E_i\). |
| 347 | Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\). |
Historical background: Also, AC for a well ordered set ofsets (40) is true, but not all cardinals are comparable with \(\aleph_1\)(41 is false) and for each \(n\ge 2\) the set of \(n\)-element subsets of \(A\)has no choice function (45(\(n\)) is false). A.~Rubin shows that each pairof cardinals has a greatest lower bound (187 is true). (See Note 108.)Truss proves that every linearly ordered set has a choice function (202 istrue). Since 202 implies \(UT(WO,WO,WO)\) (231) and in every FM model 231implies \((\forall\alpha)UT(\aleph_{\alpha},\aleph_{\alpha},\aleph_{\alpha})\),Form 23 is true. Since 202 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 70 (There is a non-trivial ultrafilter on \(\omega\).) is truein every FM model and 9 + 70 implies 63 (Every infinite set has anon-trivial ultra filter.), 63 is also true. Blass [1979] hasshown that 191 (There is a set \(X\) such that for each set \(a\) there is anordinal \(\alpha\) and a function \(f\) mapping \(X\times\alpha\) onto \(a\).) istrue in every permutation model with a set of atoms. He has also shownthat 8 (\(C(\aleph_0,\infty)\)) + 191 + 193 (Every Abelian group is ahomomorphic image of a free projective Abelian group.) iff AC. Since 202implies 8,Form 193 must be false. (See Note 60 for definitions.) Theset of all pairs of atoms is a set of well orderable sets that has nochoice function. Thus, \(C(\infty,2)\) (88) and \(KW(\infty,WO)\) (323) arefalse. Since \(C(WO,<\aleph_0)\) (122) is true (202 implies 122) it followsthat MC (67) is false because 67 + 122 implies AC. (See [67 B].) It isshown in Higasikawa [1995] that the Idemmultiple PartitionPrinciple,Form 347, is false.
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