Cohen \(\cal M23\): Hodges' Model | Back to this models page
Description: Let \(\cal M\) be a countable transitive model of \(ZFC + V = L\) and let \(\kappa\) be a regular cardinal in \(\cal M\)
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 |
|---|---|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 180 | Every Abelian group has a divisible hull. (If \(A\) and \(B\) are groups, \(B\) is a divisible hull of \(A\) means \(B\) is a divisible group, \(A\) is a subgroup of \(B\) and for every non-zero \(b \in B\), \(\exists n \in \omega \) such that \(0\neq nb\in A\).) Fuchs [1970], Theorem 24.4 p 107. |
Historical background: Let\(A=\sum_{\alpha <\kappa}A_\alpha\), where each \(A_\alpha\) is a copy of thecyclic group \(Z(4)\). Using forcing conditions of the form \(p: A\times\kappa\times\kappa\to 2\), where \(p\) is a partial function with domain ofcardinality less than \(\kappa\), it is shown in Hodges [1976b]that there is a generic extension \(\cal M[G]\) in which there is a genericcopy of \(A\). It is also shown in Hodges [1976b] that \(\cal M[G]\)can be cut down to a transitive model of ZF which contains \(A\) and allelements are definable from elements of \(\cal M\) and sequences of genericsets of length less than \(\kappa\). This is the model \(\cal M23\). It isshown in Hodges [1980] that in \(\cal M23\), \(A\) is an Abeliangroup with no divisible hull (180 is false). (If \(A\) and \(B\) are groups,``\(B\) is a divisible hull of \(A\)'' means \(B\) is a divisible group, \(A\) isa subgroup of \(B\) and for every non-zero \(b\in B\), \(\exists n\in\omega \)such that \(0\neq nb\in A\). An Abelian group \(B\) is divisible if \(n\)divides \(a\) for all \(a\in B\) and positive \(n\in\omega\).)
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