Fraenkel \(\cal N38\): Howard/Rubin Model I | Back to this models page
Description: Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering
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 |
|---|---|
| 8 | \(C(\aleph_{0},\infty)\): |
| 14 | BPI: Every Boolean algebra has a prime ideal. |
| 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 \(|\bigcup A| = \aleph _{\alpha }\). |
| 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. |
| 113 | Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact. |
| 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 \(B\). |
| 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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 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(\(n\)). |
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 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. |
| 144 | Every set is almost well orderable. |
| 152 | \(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.) |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 181 | \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function. |
| 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. |
| 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: Thatis, if \(a,b \in {\Bbb Q}^\omega\) then \(a < b\) if and only if there is some\(n \in \omega\) such that \((\forall j < n )(a_j = b_j)\) and \(a_n < b_n\).\smallskipFor each \(b \in A\) and \(n \in \omega\),\item{1.} \(A^n_b = \{ a \in A \mid a_i = b_i\hbox{ for } 0\le i\le n \,\}\) is the n--level block containing \(b\).\item{}(\(A^n_b\) will not be in the model we are defining.)\item{2.} The sequence \(\langle b_{n+1}, b_{n+2}, \ldots \rangle\) isthe position of \(b\) in its n--level block.\item{3.} \(\eusb B^n = \{ A^n_a \mid a \in A \}\) is the set of n--levelblocks.\item{4.} \(\le_n\) is the relation on \(\eusb B^n\) defined by \(A^n_a\le_n A^n_b\) if and only if \(a \le b \).\item{5.} Let \(f\) be an order automorphism of \((\eusb B^n,\le_n)\).(See lemmas A and B in Howard/Rubin [1996].) We define \(\phi_f\)to be the unique order automorphism of \((A,\le)\) which satisfies\(\phi_f''A^n_a = f(A^n_a)\) for all \(a \in A\) and such that for all \(a\inA\), \(a\) and \(\phi_f(a)\) have the same position in their \(n\)-level blocks.(By 2 above, this means that \((\forall a \in A)(\forall i > n) (a_i =\phi_f(a)_i)\).)\item{6.} For \(n \in \omega\), \(G_n\) is the group \(\{\, \phi_f : f \)is an order automorphism of \((\eusb B^n, \le_n)\,\}\).\smallskip\(\cal G\) is the group \(\bigcup_{n\in\omega} G_n\). (Note that for \(n < m\),\(G_n \subseteq G_m\).) \(S\) is the set of all \(E\subseteq A\) which satisfythe following conditions:\item{(a)} \(E\) is well ordered by the ordering \(\le\) on \(A\).\item{(b)} \((\forall n \in \omega)( E \cap A^n_a \ne\emptyset\) for only finitely many \(a \in A)\)\item{(c)} E is countable.\parIt is shown in Howard/Rubin [1996] that the Boolean Prime IdealTheorem (14) and the Axiom of Choice for a denumerable family of sets (8)are true, but both the Principle of Dependent Choices (43) and the Axiomof Choice for families of cardinality \(2^{\aleph_0}\) (181) are false. Byusing a slight modification of the proof of the Tychonoff CompactnessTheorem (see, for example, Kelley [1955], p143), it can be shownthat 14 + 8 implies that the product of a countable number of compactspaces is compact (113). It follows from Note 104, that the \(2m=m\)principle (3) is false. Using the fact that the Kinna-Wagner SelectionPrinciple (15) is false and that the Ordering Principle (30) is true (14implies 30), Krom proves that there is an ordered topological space thatis not normal (118 is false). Since 8 is true, it follows fromBrunner [1982a] that in this model there is a set that cannot bewell ordered and does not have an infinite Dedekind finite subset, (163 isfalse). (Form 8 plusForm 163 iff AC.) Since form [154 A] (If \(X\) is acompact \(T_2\) topological space, then \(X^\omega\) is compact.) is true (14implies 154) and 43 is false, 106 (the Baire Category Theorem for compact\(T_2\) spaces) is false because 106 + 154 implies 43. (See Brunner [1984b] and Note 95.) Form 14 implies 49 (Every partial ordering canbe extended to a linear ordering.) and 49 + 51 (Every linear ordering hasa cofinal sub well ordering.) iff AC. (See Morris [1969] andNote 121.) Thus,Form 51 is false. SinceForm 14 impliesForm 30 (OrderingPrinciple) andForm 113 impliesForm 296 (Every infinite set is thedisjoint union of infinitely many infinite sets.), it follows fromPincus [1997] thatForm 295 (Every infinite set has a denselinear ordering.) is true.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 proved in Howard [1973] that in every FM model 14 implies the Weak Sikorski Theorem(317), so 317 is also true.Form 40 (\(C(WO,\infty)\)) is false because 40implies 106 andForm 122 (\(C(WO,<\aleph_0)\)) is true because 60 implies122. Therefore,Form 328 (\(MC(WO,\infty)\)) is false because \(122+328\to40\).
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