Implications

This non-implication, Form 315 \( \not \Rightarrow \) Form 308-p, 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: 5463, whose string of implications is:
    218 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 25 \(\Rightarrow\) 315
  • A proven non-implication whose code is 5. In this case, it's Code 3: 544, Form 218 \( \not \Rightarrow \) Form 308-p 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 308-p <p> If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 315 \( \not \Rightarrow \) Form 308-p 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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