Fraenkel \(\cal N48\): Pincus' Model XI | Back to this models page
Description: \(\cal A=(A,<,C_0,C_1,\dots)\) is called anordered colored set (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 \(a\in C_n\)
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. |
| 14 | BPI: Every Boolean algebra has a prime ideal. |
| 16 | \(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function. |
| 23 | \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). |
| 24 | \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function. |
| 37 | Lebesgue measure is countably additive. |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 130 | \({\cal P}(\Bbb R)\) is well orderable. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 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. |
| 295 | DO: Every infinite set has a dense linear ordering. |
| 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}. |
| 317 | Weak Sikorski Theorem: If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\). |
| 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. |
| 106 | Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire. |
| 118 | Every linearly orderable topological space is normal. Birkhoff [1967], p 241. |
| 126 | \(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). |
| 131 | \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). |
| 144 | Every set is almost well orderable. |
| 253 | \L o\'s' Theorem: If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. |
| 296 | Part-\(\infty\): Every infinite set is the disjoint union of infinitely many infinite sets. |
| 328 | \(MC(WO,\infty)\): For every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.) |
Historical background: If \(a\inC_n\), \(C_n\) is called the color of \(a\). In this model we take theset of atoms to be a denumerable, universal, homogeneous OC set. (SeeJech [1973b] p101 for definitions.) It follows that \(<\) is adense unbounded linear ordering such that between any two distinctelements of \(A\) there are elements of every color. \(\cal G\) is the groupof automorphisms of \(\cal A\) and \(S\) is the set of all finite subsets of\(A\). Pincus proves that each \(C_i\), where \(i\in\omega\), is an infinite setwhich is not the union of an infinite number of pairwise disjoint infinitesets (296 is false). Pincus also proves that there is a dense linearordering on every set (295 is true) and the Boolean Prime Ideal Theorem(14) is true. Form 14 implies 49 (Every partial ordering can be extendedto a linear ordering.) and 49 + 51 (Every linear ordering has a cofinalsub well ordering.) iff AC. (See Morris [1969] and Note 121.)Thus,Form 51 is false.Form 154 (Tychonoff's compactness theorem forcountably many \(T_2\) spaces) is true (14 implies 154) and 43 is false (43implies 296). Thus, it follows from Brunner [1983c] and Note 95that 106 (the Baire category theorem for compact Hausdorff spaces) isfalse (106 + 154 iff 43).Form 14 implies the Ordering Principle (30) andthe Kinna-Wagner Principle (15) is false in every FM model. It is shown inKrom [1986] that there is an orderable topological space that isnot normal (118 is false).Form 14 implies 62 (\(C(\infty,<\aleph_0)\)) andHoward [1973] has shown that in every FM model 62 implies 60(\(C(\infty, WO)\)). Thus, 60 is true. It is also shown in Howard [1973] that in every FM model 14 implies the Weak Sikorski Theorem(317), so 317 is true.Form 30 (Ordering Principle) +Form 126(\(MC(\aleph_0,\infty)\)) impliesForm 8 (\(C(\aleph_{0},\infty)\)). Since 30is true and 8 is false, it follows that 126 is false. SinceForm 165(\(C(WO,WO)\)) is true, (60 implies 165) it follows from Note 2(8 and 9)that 16 and 24 are also true.
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