Implications

This non-implication, Form 12 \( \not \Rightarrow \) Form 380, whose code is 4, 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: 1265, whose string of implications is:
    9 \(\Rightarrow\) 376 \(\Rightarrow\) 377 \(\Rightarrow\) 378 \(\Rightarrow\) 11 \(\Rightarrow\) 12
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1435, Form 9 \( \not \Rightarrow \) Form 380 whose summary information is:
    Hypothesis Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

    Conclusion Statement
    Form 380 <p> \(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function. </p>

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

The conclusion Form 12 \( \not \Rightarrow \) Form 380 then follows.

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

Name Statement
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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