Fraenkel \(\cal N37\): A variation of Blass' model, \(\cal N28\) | Back to this models page
Description: Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)
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. |
| 144 | Every set is almost 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
| 316 | If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\) has the fixed point property if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is complete if every subset of \(A\) has a least upper bound.) |
| 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 |
|---|---|
| 51 | Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117. |
| 129 | For every infinite set \(A\), \(A\) admits a partition into sets of order type \(\omega^{*} + \omega\). (For every infinite \(A\), there is a set \(\{\langle C_j,<_j \rangle: j\in J\}\) such that \(\{C_j: j\in J\}\) is a partition of \(A\) and for each \(j\in J\), \(<_j\) is an ordering of \(C_j\) of type \(\omega^* + \omega\).) |
| 192 | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 293 | For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. |
Historical background: \(\cal G\) is the group of permutations of \(A\)generated by \(\{\phi_i: i\in\omega\}\), where \(\phi_i(a_{i,j}) = a_{i,j+1}\)and if \(k\ne i\), \(\phi_i(a_{k,j})=a_{k,j}\). \(S\) is the set of finitesubsets of \(A\). For each \(i\in\omega\), let \(E_i=\{a_{i,2j}: j\in\Bbb Z\}\)and \(O_i=\{a_{i,2j+1}: j\in\Bbb Z\}\). Then \(B=\{E_i: i\in\omega\} \cup\{O_i: i\in\omega\}\) is a partition of \(A\). We define a linear ordering\(\precsim\) on \(A\) such that \(a_{i,j}\precsim a_{k,l}\) if \(i < k\), or \(i =k\) and \(j\le l\), the lexicographical ordering. Note that \(\{a_{i,0}\}\) isa support for \(E_i\) and \(O_i\), while the empty set is a support for\((A,\precsim)\) and \(B\). Thus, \(A\) can be linearly ordered, but \(B\) cannotbe linearly ordered (293 is false). \(B\) is also an infinite set whichdoes not admit a partition into sets of order type \(^*\omega + \omega\) soform 129 is 5alse. In H\"oft/Howard [1981] it is shown that\((A,\precsim)\) has no cofinal well ordered subset (form 51 is false) andthat every linear ordering with the fixed point property is complete (316is true). (\((A,\precsim)\) has the fixed point property if everyfunction \(f:A\to A\) satisfying \((x\precsim y\Rightarrow f(x)\precsimf(y))\) has a fixed point, and (\((A,\precsim)\) is complete if everynon-empty subset of \(A\) has a least upper bound.) In Note 155 we showthat ``every set is almost well orderable'' (form 144) is true.
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