Fraenkel \(\cal N44\): Gross' model | Back to this models page
Description: \(A\) is a vector space over a finite field withbasis \(B = \bigcup_{i\in \omega} B_i\) where the \(B_i\) are pairwisedisjoint and \(|B_i| = 4\) for each \(i\in\omega\)
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. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 130 | \({\cal P}(\Bbb R)\) is 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 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. |
| 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(\(n\)). |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 236 | If \(V\) is a vector space with a basis and \(S\) is a linearly independent subset of \(V\) such that no proper extension of \(S\) is a basis for \(V\), then \(S\) is a basis for \(V\). |
Historical background: Let \(B_i =\{b_{i1},\ldots,b_{i4} \}\). \(G\) is the group of vector spaceautomorphisms \(\phi\) of \(A\) such that there is a permutation \(\pi\) of\(\omega\) and for each \(i\in\omega\) there is a permutation \(\alpha_i\) of\(\{1,2,3,4\}\) in the group generated by the permutations \((1,2)\), \((3,4)\),and \((1,3)(2,4)\) such that for all \(i\in\omega\) and \(j\in\{1,2,3,4\}\),\(\phi(b_{ij}) = b_{\pi(i)\alpha_i (j)}\). \(S\) is the set of finite subsetsof \(A\). In Gross [1975] it is shown that, in \(\cal N44\), thereis a vector space \(V\) with a basis and a subset \(T\subseteq V\) such thatno finite extension of \(T\) is a basis and no infinite extension of \(T\) isindependent. (Hence,Form 236 is false).
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