Fraenkel \(\cal N2^*(3)\): Howard's variation of \(\cal N2(3)\) | Back to this models page
Description: \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)
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 |
|---|---|
| 6 | \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable. |
| 37 | Lebesgue measure is countably additive. |
| 111 | \(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 141 | [14 P(\(n\))] with \(n = 2\): Let \(\{A(i): i\in I\}\) be a collection of sets such that \(\forall i\in I,\ |A(i)|\le 2\) and suppose \(R\) is a symmetric binary relation on \(\bigcup^{}_{i\in I} A(i)\) such that for all finite \(W\subseteq I\) there is an \(R\) consistent choice function for \(\{A(i): i \in W\}\). Then there is an \(R\) consistent choice function for \(\{A(i): i\in I\}\). |
| 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}. |
| 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 333 | \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd. |
| 361 | In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325. |
| 363 | There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325. |
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 |
|---|---|
| 110 | Every vector space over \(\Bbb Q\) has a basis. |
| 154 | Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact. |
| 164 | Every non-well-orderable set has an infinite subset with a Dedekind finite power set. |
| 288-n | If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. |
| 334 | \(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of sets such that for all \(x\in X\), \(|x|\ge 2\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is even. |
| 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 \(E_i\). |
| 346 | If \(V\) is a vector space without a finite basis then \(V\) contains an infinite, well ordered, linearly independent subset. |
| 358 | \(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
Historical background: For each \(i\in\omega\) define a function \(\eta_i: T_i\to T_i\) suchthat \(\eta_i: a_i\mapsto b_i\mapsto c_i\). \(\cal G\) is the group of allpermutations \(\phi\) of \(A\) such that for each \(i\in\omega\), \(\phi|T_i\) iseither the identity, \(\eta_i\), or \(\eta_i^2\). \(S\) is the set of all finitesubsets of \(A\). In this model, \(B\) is a denumerable set of triples thathas no choice function and has no Kinna-Wagner selection function so\((\forall n\in\omega-\{0,1\})C(\aleph_0,n)\) (288) is false and\(KW(\aleph_0,<\aleph_0)\) (358) is false. (Since the Boolean Prime IdealTheorem (14) implies 288,Form 14 is also false.) Howard shows that if \(P\)is any set of pairs in the model and \(R\) is a symmetric relation on\(\bigcup P\) such that every finite subset of \(P\) has an \(R\)-consistentchoice function, (A choice function \(f\) on \(X\) is \(R\)-consistent if forall \(u, v\in X\), \(u\not= v\) implies \(f(u) R f(v)\).) then \(P\) has an\(R\)-consistent choice function (141 is true). It is shown in Note 93 that\(MC(\infty,\infty,\hbox{ odd})\) (333) is true. Keremedis [1996a]proves that \(333 + 334 (MC(\infty,\infty,\hbox { even})\) iff AC, so 334 isfalse. \(\cal N2^*(3)\) can be extended to \(\cal N2^*(k)\) where \(k\in\omega-\{0,1,2\}\), \(k\geq 3\). Then \(T_i\) has \(k\) elements, the permutations arecircular, and supports are finite. It is shown in De la Cruz, Hall,Howard, Keremedis, Rubin [2002b] that for each \(n\in \omega -\{0,1\}\) there is a \(k\in \omega - \{0,1,2\}\) such that 422(\(n\)) is truein \(\cal N2^*(k)\) and 423 is false. Form 110 and 346 are false in thismodel since it is shown in Keremedis [2001a], theorem 7 thateach of forms 110 and 346 implyForm 373(\(n\)) for all \(n\in \Bbb N - \{0,1\}\) andForm 373(3) is clearly false in \(\cal N2^*(3)\) .
Back