Cohen \(\cal M37\): Monro's Model III | Back to this models page
Description: This is a generic extension of \(\cal M1\) in which there is an amorphous set (Form 64 is false) and \(C(\infty,2)\) (Form 88) is false
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 |
|---|---|
| 371 | There is an infinite, compact, Hausdorff, extremally disconnected topological space. Morillon [1993]. |
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 |
|---|---|
| 64 | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 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. |
Historical background: The model is constructed as follows: If \(b\) is the countable set ofgeneric reals from which \(\cal M1\) is constructed, define a partialordering \((Q,\le)\) in \(\cal M1\) where \(Q\) is the set of ordered pair\((A,\rho)\) such that \(A\) is a finite subset of \(b\) and \(\rho:A^2\to\{0,1\}\) satisfies \(\{ (a,b) : \rho(a,b) =1\,\}\) is an equivalencerelation on \(A\). The ordering \(\le\) is defined by \((B,\sigma)\le(A,\rho)\)if and only if \(A\subseteq B\) and \(\sigma\) extends \(\rho\). Let \(F\) be \(Q\)generic over \(\cal M1\). Then Monro's model, \(\cal M37\), is \(\cal M1[F]\).In Morillon [1993] it is shown that ifForm 64 is false thenthere is an infinite, compact, Hausdorff, extremally disconnectedtopological space and therefore 371 is true.
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