Implications

This non-implication, Form 111 \( \not \Rightarrow \) Form 411, 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: 9929, whose string of implications is:
    250 \(\Rightarrow\) 111
  • 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: 9576, whose string of implications is:
    411 \(\Rightarrow\) 412 \(\Rightarrow\) 10 \(\Rightarrow\) 249

The conclusion Form 111 \( \not \Rightarrow \) Form 411 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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