Cohen \(\cal M22\): Plotkin's Model I | Back to this models page
Description: Let \(T\) be a complete first order theory with equality which has infinite models and is \(\aleph_0\)-categorical
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. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 176 | Every infinite, locally finite group has an infinite Abelian subgroup. (Locally finite means every finite subset generates a finite subgroup.) |
Historical background: Let\(\cal M\) be a countable standard transitive model of ZFC which contains acountable model \(\frak A\) of \(T\). It is shown in Plotkin [1969]that there is a generic extension of \(\cal M\) that contains an isomorphiccopy, \(\frak A'\), of \(\frak A\), such that in the symmetric submodelconstructed, \(\frak A'\) is infinite, Dedekind finite, and the relations of\(T\) are definable. Let \(p\) be an odd prime and let \(T\) be the first ordertheory of an infinite extra-special \(p\)-group of exponent \(p\). (See note24 for definitions.) It is shown in Felgner [1975] that thistheory is complete and \(\aleph_0\)-categorical. \(\cal M22\) is theresulting model. Plotkin [1981] proves the countableextra-special \(p\)-groups of exponent \(p\) in \(\cal M22\) are all locallyfinite and have no infinite Abelian subgroups. (Locally finite means thatevery finite subset generates a finite subgroup.) Thus,Form 176 isfalse. (Since the groups constructed are infinite and Dedekind finite,form 9 is also false.)
Back