Cohen \(\cal M1\): Cohen's original model | Back to this models page
Description: Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them
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 |
|---|---|
| 14 | BPI: Every Boolean algebra has a prime ideal. |
| 15 | \(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)). |
| 31 | \(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. |
| 60 |
\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function. |
| 82 | \(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) |
| 118 | Every linearly orderable topological space is normal. Birkhoff [1967], p 241. |
| 128 | Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 165 | \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. |
| 170 | \(\aleph_{1}\le 2^{\aleph_{0}}\). |
| 191 | \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). |
| 277 | \(E(D,VII)\): Every non-well-orderable cardinal is decomposable. |
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 |
|---|---|
| 13 | Every Dedekind finite subset of \({\Bbb R}\) is finite. |
| 17 | Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20. |
| 50 | Sikorski's Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141. |
| 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. |
| 106 | Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire. |
| 131 | \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). |
| 144 | Every set is almost well orderable. |
| 164 | Every non-well-orderable set has an infinite subset with a Dedekind finite power set. |
| 253 | \L o\'s' Theorem: If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. |
| 289 | If \(S\) is a set of subsets of a countable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element. |
| 299 | Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces. |
| 300 | Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. |
| 369 | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
Historical background: It follows from the work of Halpern and Levy that the Boolean Prime Ideal Theorem (BPI,14) and the Kinna-Wagner Selection Principle (15) are true in this model. Cohen proved that in this model there is an infinite subset of the reals that has no denumerable subset (13 is false). Also, the Krein-Milman Theorem (KM, 65) is false because BPI + KM \(\to\) AC and Haddad and Morillon proved that in \(\mathcal M1\) every orderable topological space is normal (118 is true). Jech has shown that the Axiom of Choice for a set of well orderable sets (\(C(\infty, WO)\), 60)is true and the countable union theorem (\(UT(\aleph_0,\aleph_0,\aleph_0)\),31) is true. Blass proved that Ramsey's Theorem (17) is false. Brunner [1982a] has shown that every set that cannot be well ordered has an infinite Dedekind finite subset (163 is true), but the interval \([0,1]\) of real numbers is not well orderable and has no infinite subset with a Dedekind finite power set (164 is false). It is shown in Note 102 that there is a set \(S\) consisting of finite unions of sets in \(b\), the set of generic reals, that is closed with respect to chain unions, but has no \( \subseteq\)-maximal element (289 is false). Brunner [1984f] shows that in this model on every infinite set there is a Hausdorff topology with an infinite set of non-isolated points (128 is true). Blass [1979] has shown that in every symmetric model there is a set \(X\) such that for each set \(a\), there is an ordinal \(\alpha\) and a function \(f\) mapping \(X\times\alpha\) onto \(a\) (191 istrue). (See Note 59.) Since BPI is true, it follows from Howard [1975] that Łos' Theorem (253) is false. Monro proves that every cardinal that cannot be well ordered is decomposable (i.e. is the sum of two smaller positive cardinals), so 277 is true. Since \(\mathbb R\) cannot be well ordered in this model and Form 277 is true, it follows that 369 (\(\mathbb R\) is not decomposable) is false. In Jech [1973b] (prob5.20), it is shown that the power set of an infinite set is Dedekind infinite (82 is true). David shows that no infinite subset of \(b\) is almost well orderable so 144 is false. See Note 25 for definitions. By the results of Bell [1983], \cite{1988} and Monro [1972] the following three forms are false in \(\mathcal M1\): Form 50 (Sikorski's extension theorem on Boolean homomorphisms), Form 299 (Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces.) and Form 300 (Any continuous surjection between extremally disconnected Hausdorff spaces has an irreducible restriction to a closed subset of its domain.) See notes 60 and 114 for definitions. Since 154 is true (14 implies 154) and 43 is false (43 implies 13), Form 106 (the Baire Category Theorem for compact \(T_2\) spaces) is false because 106 + 154 implies 43. (See Brunner [1984b] and Note 95.) In this model, \(\mathbb R\) with the order topology is a separable metric space that is not Lindel\"of ([94 I] is false). Also, there is a dense set \(K\subseteq \mathbb R\) which has no denumerable subset such that \(K\) is a second countable metric space that is neither Lindel\"of nor separable which shows that [94 J] and [94 K] are false. (See Good/Tree [1995].) Howard/Keremedis/Rubin/Stanley/Tachtsis [1999] have shown that form 369 (If \(\mathbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\).) is false and that Form 170 (\(\aleph_1\leq2^{\aleph_0}\)) is true.
Back