Implications

This non-implication, Form 374-n \( \not \Rightarrow \) Form 249, 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: 3987, whose string of implications is:
    250 \(\Rightarrow\) 47-n \(\Rightarrow\) 423 \(\Rightarrow\) 374-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 252, Form 250 \( \not \Rightarrow \) Form 249 whose summary information is:
    Hypothesis Statement
    Form 250 <p> \((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function. </p>

    Conclusion Statement
    Form 249 <p> If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch. </p>

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

The conclusion Form 374-n \( \not \Rightarrow \) Form 249 then follows.

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

Name Statement
\(\cal N35\) Truss' Model IV The set of atoms, \(A\), is denumerable andeach element of \(A\) is associated with a finite sequence of zeros andones

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