Cohen \(\cal M4\): Pincus' Model I | Back to this models page
Description: This model has many of the properties of the model given in Cohen [1966] (p 143)
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 |
|---|---|
| 52 | Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\). |
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 |
|---|---|
| 65 | The Krein-Milman Theorem: Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in \(K\).) Rubin, H./Rubin, J. [1985] p. 177. |
| 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. |
| 358 | \(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
Historical background: The ground model, \(\cal M\),is a model of ZF + V = L. Forcing conditions are functions from countablesubsets of \(\omega_1\) to \(\{0,1\}\). \(\cal M4\) is a symmetric submodel ofthe corresponding generic extension of \(\cal M\). Pincus proves that theHahn-Banach Theorem (52) is true, but there is an infinite set with nodenumerable subset (9 is false), and there is a denumerable family ofpairs that has no choice function. It follows that\(KW(\aleph_0,<\aleph_0)\) (358) and \(C(\infty,2)\) (88) are false. Pincusalso proves that the Krein-Milman Theorem (65) is false. (Since BPI (form14) implies 358, BPI is also false.)
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