This non-implication,
Form 48-K \( \not \Rightarrow \)
Form 76,
whose code is 4, is constructed around a proven non-implication as follows:
Hypothesis | Statement |
---|---|
Form 60 | <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </p> |
Conclusion | Statement |
---|---|
Form 76 | <p> \(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every 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 \(x\). </p> |
The conclusion Form 48-K \( \not \Rightarrow \) Form 76 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
Name | Statement |
---|---|
\(\cal M1\) Cohen's original model | Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them |
\(\cal M20\) Felgner's Model I | Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\) |
\(\cal N3\) Mostowski's Linearly Ordered Model | \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\) |
\(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) | \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\) |
\(\cal N29\) Dawson/Howard Model | Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\) |
\(\cal N48\) Pincus' Model XI | \(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\) |