This non-implication,
Form 283 \( \not \Rightarrow \)
Form 296,
whose code is 6,
is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the
transferability criterion. Click
Transfer details for all the details)
| Hypothesis | Statement |
|---|---|
| Form 317 | <p> <strong>Weak Sikorski Theorem:</strong> If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\). |
| Conclusion | Statement |
|---|---|
| Form 296 | <p> <strong>Part-\(\infty\):</strong> Every infinite set is the disjoint union of infinitely many infinite sets. </p> |
The conclusion Form 283 \( \not \Rightarrow \) Form 296 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N3\) 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 \(A\) |
| \(\cal N48\) 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 \(a\in C_n\) |