Cohen \(\cal M33\): Plotkin's Model II | Back to this models page
Description: The construction is similar to the construction of \(\cal M22\)
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 |
|---|---|
| 9 | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 88 | \(C(\infty ,2)\): Every family of pairs has a choice function. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 247 | Every atomless Boolean algebra is Dedekind infinite. \ac{Plotkin} \cite{1976}, notes 86 and 94. |
| 248 | For any \(\kappa\), \(\kappa\) is the cardinal number of an infinite complete Boolean algebra if and only if \(\kappa^{\aleph_0} = \kappa \). |
Historical background: An atomless Boolean algebra in a countablestandard model \(\cal M\) of ZF + V = L is embedded in a generic extension,\(\cal M33\), of \(\cal M\) in such a way that the isomorphic copy, \(B\), ofthe Boolean algebra has the property that it is infinite, atomless, andDedekind finite (9 and 247 are false) and \(|B|^{\aleph_0}> |B|\) (248 is false). Plotkin also proves that \(C(\infty,2)\) (88) isfalse in this model. Additional properties of \(B\) include the following:Every filter in \(B\) is principal; \(B\) has no infinite free subalgebra; \(B\)is not the direct union of complete weakly homogeneous Boolean algebras;\(B\) is not inversely reducible. See Plotkin [1976] fordefinitions.
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