Fraenkel \(\cal N20\): Truss' Model II | Back to this models page
Description:
Let \(X=\{a(i,k,l): i\in 2, k\in \Bbb Z, l\in\omega\}\), \(Y=\{a(i,j,k,l): i,j\in 2, k\in\Bbb Z, i\in\omega\}\) and \(A\) is the disjoint union of \(X\) and \(Y\)
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(\(n\)). |
| 136-k | Surjective Cardinal Cancellation (depends on \(k\in\omega-\{0\}\)): For all cardinals \(x\) and \(y\), \(kx\le^* ky\) implies \(x\le^* y\). |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
Historical background: The set of atoms is countable withorder type \((2\times\Bbb Z\times\omega)\cup(2^2\times\Bbb Z\times\omega)\).Let \(V(k,l)=\{a(i,k,l): i\in 2\}\), \(W(k,l)=\{a(i,j,k,l): i,j\in 2\}\),\(S=\{V(k,l): k\in\Bbb Z, l\in\omega\}\), \(T=\{W(k,l): k\in\Bbb Z,l\in\omega\}\), and \(U=\{\{V(k,l),W(k,l)\}: k\in\Bbb Z, l\in\omega\}\cup\{\{V(k+1,l),W(k,l)\}: k\in\Bbb Z, l\in\omega\}\). Using forcing, constructa generic choice function \(F\) on the set of all pairs of elements in\(S\cup T\). Using \(F\), it is possible to construct a function \(f\) that maps\(2\times Y\) onto \(2\times X\). \(\cal G\) is the group of all permutationsof \(A\) that leaves \(f\), \(X\), \(Y\), \(S\), \(T\), and \(U\) fixed. The set ofsupports is the finite subsets of \(A\). (This is a Fraenkel-Mostowski-Cohenmodel.) In this model, there is a mapping of \(2\times Y\) onto \(2\times X\),but no mapping of \(Y\) onto \(X\).
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