This non-implication,
Form 63 \( \not \Rightarrow \)
Form 107,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 63 | <p> \(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. <br /> <a href="/books/8">Jech [1973b]</a>, p 172 prob 8.5. </p> |
| Conclusion | Statement |
|---|---|
| Form 88 | <p> \(C(\infty ,2)\): Every family of pairs has a choice function. </p> |
The conclusion Form 63 \( \not \Rightarrow \) Form 107 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N1\) The Basic Fraenkel Model | The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\) |
| \(\cal N2(\aleph_{\alpha})\) Jech's Model | This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\) |
| \(\cal N9\) Halpern/Howard Model | \(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\) |
| \(\cal N10\) Höft/Howard/Mostowski Model | (The model is a variation of\(\cal N3\).) \(A\) as ordered by \(\precsim\) has the same order type as therationals; \(\cal G\) is the group of all order automorphisms of \(A\); \(S\) isthe set of all subsets \(E\) of \(A\) that satisfy the following threeconditions:\item{1.} \(E\) is well ordered by \(\precsim\).\item{2.} \(E\) is bounded in \(A\).\item{3.} If \(b:\alpha\to E\) is an order preserving bijection from\(\alpha\) onto \(E\) and if \(\lambda < \alpha\) is a limit ordinal then\(\{b(\beta) : \beta < \gamma\}\) has no least upper bound in\((A,\precsim)\).\par\noindentIn <a href="/articles/H\"oft/Howard-1994">H\"oft/Howard [1994]</a> it is shown that, in \(\cal N10\), everyDedekind finite set is finite (9 is true), but \((A,\precsim)\) is alinearly ordered set with no infinite descending sequences that cannot bewell ordered (77 is false) |
| \(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) | 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 |
| \(\cal N23\) Howard/Mostowski Linearly Ordered Model | This model is amodification of \(\cal N3\) |
| \(\cal N24\) Hickman's Model I | This model is a variation of \(\cal N2\) |
| \(\cal N24(n)\) An extension of \(\cal N24\) to \(n\)-element sets, \(n>1\).\(A=\bigcup B\), where \( B=\{b_i: i\in\omega\}\) is a pairwise disjoint setof \(n\)-element sets | \(\cal G\) is the group of all permutations of \(A\)which are permutations of \(B\); and \(S\) is the set of all finite subsets of\(A\) |
| \(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\) | The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\) |
| \(\cal N41\) Another variation of \(\cal N3\) | \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\) |
| \(\cal N53\) Good/Tree/Watson Model I | Let \(A=\bigcup \{Q_n:\ n\in\omega\}\), where \(Q_n=\{a_{n,q}:q\in \Bbb{Q}\}\) |