Implications

This non-implication, Form 250 \( \not \Rightarrow \) Form 338, whose code is 4, is constructed around a proven non-implication as follows:

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

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2437, whose string of implications is:
    338 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 249

The conclusion Form 250 \( \not \Rightarrow \) Form 338 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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