Implications

This non-implication, Form 116 \( \not \Rightarrow \) Form 71-alpha, 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: 9933, whose string of implications is:
    67 \(\Rightarrow\) 116
  • A proven non-implication whose code is 5. In this case, it's Code 3: 165, Form 67 \( \not \Rightarrow \) Form 373-n whose summary information is:
    Hypothesis Statement
    Form 67 <p> \(MC(\infty,\infty)\) \((MC)\), <strong>The Axiom of Multiple Choice:</strong> For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). </p>

    Conclusion Statement
    Form 373-n <p> (For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1079, whose string of implications is:
    71-alpha \(\Rightarrow\) 9 \(\Rightarrow\) 10 \(\Rightarrow\) 288-n \(\Rightarrow\) 373-n

The conclusion Form 116 \( \not \Rightarrow \) Form 71-alpha then follows.

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

Name Statement
\(\cal N2(n)\) A generalization of \(\cal N2\) This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\)

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