Cohen \(\cal M6\): Sageev's Model I | Back to this models page
Description: Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)
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 |
|---|---|
| 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
| 30 | Ordering Principle: Every set can be linearly ordered. |
| 34 | \(\aleph_{1}\) is regular. |
| 170 | \(\aleph_{1}\le 2^{\aleph_{0}}\). |
| 295 | DO: Every infinite set has a dense linear ordering. |
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 |
|---|---|
| 5 | \(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function. |
| 126 | \(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 330 | \(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.) |
Historical background: He proves that in \(\cal M6\),every set can be linearly ordered (30 is true) and the \(2m=m\) principle(3) holds, but the Axiom of Choice for a denumerable set of denumerablesubsets of the reals (5) is false. Gonzalez has shown that every infiniteset has a dense linear ordering in \(\cal M6\) (295 is true).Form 3 impliesform 10 (\(C(\aleph_0,<\aleph_0)\)) andForm 231 (\(UT(WO,WO,WO)\)) impliesform 5. Thus,Form 10 is true andForm 231 is false. It is shown in note123 that 10 + 163 (Every non well orderable set has an infinite Dedekindfinite subset.) implies 231. Consequently,Form 163 is false.(Halpern/Howard [1970] have shown using a permutation model thatthe \(2m=m\) principle does not imply the Axiom of Choice in ZF\(^0\). See\(\cal N9\).)Form 165 (\(C(WO,WO)\)) is false because 165 implies 5 and form122 (\(C(WO,<\aleph_0)\)) is true because 295 implies 122. Therefore, form330 (\(MC(WO,WO)\)) is false because \(122 + 330 \to 165\). In addition, 126(\(MC(\aleph_0,\infty)\)) is false because 30 is true and 5 is false. (8(\(C(\aleph_0,\infty)\)) implies 5 and 30 + 126 implies 8.) Sageev provesthat alephs are preserved in this model so, sinceForm 170 (\(\aleph_1 \leq2^{\aleph_0}\) is true in the ground model, it is true in \(\cal M6\). Italso follows that 34 (\(\aleph_1\) is regular) is true. (SeeHoward/Keremedis/Rubin/Stanley/Tachtsis [1999] Lemma 5.)
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