Cohen \(\cal M16(n)\): Monro's Model I | Back to this models page
Description: Monro has constructed an \(\omega\)-sequence of models of \(ZF\) such that,
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 |
|---|---|
| 0 | \(0 = 0\). |
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(\(n\)). |
| 81-n | (For \(n\in\omega\)) \(K(n)\): For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow {\cal P}^{n}(\alpha)\). (\({\cal P}^{0}(X) = X\) and \({\cal P}^{n+1}(X) = {\cal P}({\cal P}^{n}(X))\). (\(K(0)\) is equivalent to [1 AC] and \(K(1)\) is equivalent to the selection principle (Form 15)). |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
Historical background: Let \(H_\kappa(A,B)\) be the set of all functions\(f\subseteq A\times B\) such that \(|\hbox{dom}(f)|<\kappa\). Let \(\cal M^0=\calM\) and \(J^0=\omega\). Then \(\cal M^{k+1}\) is obtained from \(\cal M^k\) asfollows. Let \( G^{k+1}\) be \(H_\omega(J^k\times J^k,2)\)-generic over \(\calM_k\). For \(r\in J^k\), let \(G^{k+1}_r = \{(s,a): (r,s,a)\in\bigcup(G^{k+1}\}\). (\(G^{k+1}_r : J^k\to 2\).) Then \(J^{k+1}=\{G^{k+1}_r :r\in J^k\}\) and \(\cal M^{k+1}=\cal M^k[G^{k+1}]\). Let \(K(n)\) be thestatement: For every set \(x\) there is an ordinal \(\alpha\) and a 1--1 function\(f: x\longrightarrow \cal P^n(\alpha)\). (\(K(n)\) isForm 81(\(n\)); \(K(0)\) isequivalent to AC; and \(K(1)\) is equivalent to the Kinna-Wagner SelectionPrinciple,Form 15.) Monro proves that if \(\cal M_1\) and \(\cal M_2\) are twomodels such that for each \(\alpha\), \(\cal P^n(\alpha)\) is the same in eachmodel and, in addition, \(\cal M_1\ne\cal M_2\), then \(K(n-1)\) is false in eachmodel. Since, each successive pair of models in the sequence constructed hasthe property (\(*\)), it follows that for each \(n\in\omega\), there is a modelof ZF in which 81(\(n\)) is false. Monro also proves that for each\(n\in\omega\), {}\(\cal M^n\models K(n+1)\). Consequently, it follows that forall \(n > 0\), \(K(n+1)\not\to K(n-1)\).
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