Cohen \(\cal M47(n,M)\): Pincus' Model IX | Back to this models page
Description: This is the model of Pincus [1977a], Theorem 2.1 \((E)\)
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 |
|---|---|
| 43 | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 46-K | If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function. |
| 63 |
\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
|
| 214 | \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\). |
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 |
|---|---|
| 46-K | If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function. |
| 47-n | If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 329 | \(MC(\infty,WO)\): For every set \(M\) 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.) |
| 344 | If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). |
Historical background: (E). In Pincus [1977a] it is shownthat \(\cal N2(n,M )\) may be transferred to a model of ZF, \(\cal N2(n,M)T\),in which 46(\(\neg M\)) is true, but 46(\(n\in K\)) and 47(\(n\)) are false.Then \(\cal N2(n,M)T\) can be modified so that, in addition, the Principleof Dependent Choices (43) is true and for each set \(x\) of infinite sets,there is a function \(f\) such that for each \(y\in x\), \(f(y)\subseteq y\) and\(|f(y)|=\aleph_0\) (214) is true. We shall call the modified model \(\calM47(n,M)\). Since 43 implies 9 (Every Dedekind finite set is finite.) and70 (There is a non-trivial ultra filter on \(\omega\).) is true in every FMmodel, it follows that 63 (Every infinite set has a non-trivialultrafilter.) is also true. SinceForm 64 (There is no amorphous set.) istrue (30 implies 64),Form 133 (Every set is either well orderable or hasan infinite amorphous subset.) is false. In this modelForm 10(\(C(\aleph_0,<\aleph_0)\)) is true (30 implies 10) andForm 165(\(C(WO,WO)\)) is false (165 implies 32). Thus,Form 163 (Every non wellorderable set has an infinite Dedekind finite subset.) is false becauseBrunner [1982a] shows that 10 + 163 implies 165.
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