Implications

This non-implication, Form 268 \( \not \Rightarrow \) Form 7, 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)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4315, whose string of implications is:
    218 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 268
  • A proven non-implication whose code is 5. In this case, it's Code 3: 540, Form 218 \( \not \Rightarrow \) Form 10 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 10 <p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 254, whose string of implications is:
    7 \(\Rightarrow\) 9 \(\Rightarrow\) 10

The conclusion Form 268 \( \not \Rightarrow \) Form 7 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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