This non-implication,
Form 191 \( \not \Rightarrow \)
Form 255,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 191 | <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p> |
| Conclusion | Statement |
|---|---|
| Form 106 | <p> <strong>Baire Category Theorem for Compact Hausdorff Spaces:</strong> Every compact Hausdorff space is Baire. <p> |
The conclusion Form 191 \( \not \Rightarrow \) Form 255 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M1\) Cohen's original model | Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them |
| \(\cal N1\) 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 \(A\) |
| \(\cal N3\) Mostowski's Linearly Ordered Model | \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\) |
| \(\cal N4\) The Mathias/Pincus Model I | \(A\) is countably infinite;\(\precsim\) is a universal homogeneous partial ordering on \(A\) (See<a href="/articles/Jech-1973b">Jech [1973b]</a> p 101 for definitions.); \(\cal G\) is the group ofall order automorphisms on \((A,\precsim)\); and \(S\) is the set of allfinite subsets of \(A\) |
| \(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) | \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\) |
| \(\cal N8\) L\"auchli's Model II | \(A\) is countably infinite, ordered likethe rational numbers; \(\cal G\) is the group of all order automorphisms of\(A\); and \(S\) is the set of all subsets \(E\) of \(A\) such that \(E\) has atmost a finite number of accumulation points and every infinite subset of\(E\) has an accumulation point |
| \(\cal N10\) 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) |
| \(\cal N16\) Jech/Levy/Pincus Model | \(A\) has cardinality \(\aleph_{\omega}\);\(\cal G\) is the group of all permutations on \(A\); and \(S\) is the set ofall subsets of \(A\) of cardinality less that \(\aleph_{\omega}\) |
| \(\cal N16(\aleph_{\gamma})\) Levy's Model II | This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\) |
| \(\cal N17\) Brunner/Howard Model II | \(A=\{a_{\alpha,i}:\alpha\in\omega_1\,\wedge i\in\omega\}\) |
| \(\cal N18\) Howard's Model I | Let \(B= {B_n: n\in\omega}\) where the \(B_n\)'sare pairwise disjoint and each is countably infinite and let \(A=\bigcup B\) |
| \(\cal N21(\aleph_{\alpha+1})\) Jensen's Model | We assume \(\aleph_{\alpha+1}\) is a regular cardinal |
| \(\cal N24\) 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 \(A\) |
| \(\cal N29\) Dawson/Howard Model | Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\) |
| \(\cal N38\) Howard/Rubin Model I | Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering |
| \(\cal N39\) Howard's Model II | \(A\) is denumerable and is a disjoint union\(\bigcup_{i\in\omega}B_i\cup\bigcup_{i\in\omega}C_i\), where for all\(i\in\omega, |B_i|=|C_i|=\aleph_0\) |
| \(\cal N41\) 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 \(\le_n\) |
| \(\cal N43\) Brunner's Model II | The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint |
| \(\cal N46\) Höft/Howard Model I | The set \(A\) of atoms is denumerable andordered by \(\le\) so that \((A,\le)\) is order isomorphic to the rationals.\(A\) is written as the union \(A = D_1 \cup D_2 \cup D_3\) of three densedisjoint subsets |
| \(\cal N48\) Pincus' Model XI | \(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\) |
| \(\cal N49\) De la Cruz/Di Prisco Model | Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\) |
| \(\cal N55\) Keremedis/Tachtsis Model: The set of atoms \(A=\bigcup \{A_n: n\in \omega\}\), where \(A_n=\{a_{n,x}: x\in B(0,\frac1n)\}\) and \(B(0,\frac1n)= \{x: \rho(x,0)=\frac1n\}\), where \(\rho\) is the Euclidean metric | The group of permutations \(\cal G\), is the group of all rotations of the \(A_n\) through an angle \(\theta\in [0,2\pi)\), and supports are finite |