Implications

This non-implication, Form 125 \( \not \Rightarrow \) Form 76, 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: 9639, whose string of implications is:
    144 \(\Rightarrow\) 125
  • A proven non-implication whose code is 5. In this case, it's Code 3: 409, Form 144 \( \not \Rightarrow \) Form 131 whose summary information is:
    Hypothesis Statement
    Form 144 <p> Every set is almost well orderable. </p>

    Conclusion Statement
    Form 131 <p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10257, whose string of implications is:
    76 \(\Rightarrow\) 131

The conclusion Form 125 \( \not \Rightarrow \) Form 76 then follows.

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

Name Statement
\(\cal N17\) Brunner/Howard Model II \(A=\{a_{\alpha,i}:\alpha\in\omega_1\,\wedge i\in\omega\}\)
\(\cal N18\) Howard's Model I Let \(B= {B_n: n\in\omega}\) where the \(B_n\)'sare pairwise disjoint and each is countably infinite and let \(A=\bigcup B\)

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