Implications

This non-implication, Form 112 \( \not \Rightarrow \) Form 325, whose code is 6, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5240, whose string of implications is:
    218 \(\Rightarrow\) 67 \(\Rightarrow\) 112
  • A proven non-implication whose code is 5. In this case, it's Code 3: 549, Form 218 \( \not \Rightarrow \) Form 358 whose summary information is:
    Hypothesis Statement
    Form 218 <p> \((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then  there  is  a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\). </p>

    Conclusion Statement
    Form 358 <p> \(KW(\aleph_0,<\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1728, whose string of implications is:
    325 \(\Rightarrow\) 17 \(\Rightarrow\) 132 \(\Rightarrow\) 10 \(\Rightarrow\) 358

The conclusion Form 112 \( \not \Rightarrow \) Form 325 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N6\) Levy's Model I \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th prime

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