Implications

This non-implication, Form 401 \( \not \Rightarrow \) Form 332, 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: 10043, whose string of implications is:
    121 \(\Rightarrow\) 401
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1049, Form 121 \( \not \Rightarrow \) Form 132 whose summary information is:
    Hypothesis Statement
    Form 121 <p> \(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function. </p>

    Conclusion Statement
    Form 132 <p> \(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9166, whose string of implications is:
    332 \(\Rightarrow\) 343 \(\Rightarrow\) 62 \(\Rightarrow\) 132

The conclusion Form 401 \( \not \Rightarrow \) Form 332 then follows.

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

Name Statement
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)

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