Cohen \(\cal M39(\kappa,\lambda)\): Kanovei's model II | Back to this models page
Description: This model depends on the two cardinals \(\kappa < \lambda\) such that both \(\kappa\) and \(\lambda\) have cofinality \(>\omega\) and neither \(\kappa\) nor \(\lambda\) can be written as \(\theta^+\) where \(\theta\) is a cardinal of countable cofinality and such that \(\aleph_2 \le\kappa\)
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 |
|---|---|
| 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. |
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 |
|---|---|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 307 | If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\). |
Historical background: \par Suppose \(\kappa\) and \(\lambda\)are cardinals in a model \(\cal M\) of ZFC + V=L. Assume that \(\kappa\) and\(\lambda\) satisfy the properties of the preceding paragraph in \(\cal M\).Let \(P = \{\, p : p\) is a function such that dom\((p)\subseteq\lambda\times \omega\) is finite and ran\((p) \subseteq \{0,1\}\,\}\) and let\(G\) be \(P\) generic over \(M\). For each \(\zeta < \lambda\), let \(a_\zeta(k) =i\) if and only if \((\exists p\in G) (p(\zeta,k) = i)\). \(a_\zeta\) is afunction from \(\omega\) to \(\{0,1\}\) which we identify with a real numberbetween 0 and 1 whose binary expansion has digits \(a_\zeta(0), a_\zeta(1),a_\zeta(2), \ldots\) and use \([a_\zeta]\) to denote the Vitali equivalenceclass of this real number. (Vitali equivalence classes are equivalenceclasses or real numbers under the relation \(x\equiv y \leftrightarrow(\exists q\in {\Bbb Q})(x-y = q)\).) Now define the following\item{1.} \(B = \{\, b\in \cal M : b\) is a bijection from \(\lambda\) onto\(\lambda\,\}\).\item{2.} \(Z = \{\,z\subseteq \lambda\times\omega : z\) is finite\(\,\}\)\item{3.} For \(b\in B\), \(z\in Z\) and \(\zeta < \lambda\)\(za_\zeta(k) = \cases a_\zeta(k) &\hbox{for }(\zeta,k)\notin z\\ 1 - a_\zeta(k) &\hbox{for }(\zeta,k)\in z\endcases\)\item{4.} \(zb\mid u = \langle za_{b(\zeta)} : \zeta \in u \rangle\)\item{5.} \(|zb\mid u| = \langle |za_{b(\zeta)}| : \zeta \in u \rangle\)\item{6.} \(C_\theta = \{\,u\subseteq \lambda : u \in \cal M\) hascardinality \(<\theta\) in \(\cal M\,\}\)\item{7.} \(W_\kappa = \{ \, zb\mid u : z\in Z \land b\in B\land u\inC_\kappa \,\}\)\item{8.} \([W_\lambda] = \{\, |zb\mid u| : z\in Z \land b\in B \land u\inC_ \lambda\,\}\)\smallskipThe model \(\cal M39(\kappa,\lambda)\) is HOD\((W_\kappa \cup [W_\lambda]\cup \{W_\kappa,[W_\lambda]\})\). In Kanovei [1991] it is shownthat the Principle of Dependent Choices (43) is true and that\(H(2^{\aleph_0}) < H(m)\) where \(m\) is the cardinality of the set of Vitaliequivalence classes and \(H\) is Hartogs' function (307 is false). (\(H(x)\)is the smallest ordinal that is not similar to a subset of \(x\).) Since 43implies 8 (\(C(\aleph_0,\infty)\)), it follows from Brunner [1982a] that in this model there is a set that cannot be well orderedand does not have an infinite Dedekind finite subset, (163 is false).(Form 8 plusForm 163 iff AC.)
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