This non-implication,
Form 43 \( \not \Rightarrow \)
Form 409,
whose code is 4, is constructed around a proven non-implication as follows:
Hypothesis | Statement |
---|---|
Form 43 | <p> \(DC(\omega)\) (DC), <strong>Principle of Dependent Choices:</strong> 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 <a href="/articles/Tarski-1948">Tarski [1948]</a>, p 96, <a href="/articles/Levy-1964">Levy [1964]</a>, p. 136. </p> |
Conclusion | Statement |
---|---|
Form 46-K | <p> 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. </p> |
The conclusion Form 43 \( \not \Rightarrow \) Form 409 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
Name | Statement |
---|---|
\(\cal M47(n,M)\) Pincus' Model IX | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\) |