Implications

This non-implication, Form 249 \( \not \Rightarrow \) Form 378, 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: 2426, whose string of implications is:
    165 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 249
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1050, Form 165 \( \not \Rightarrow \) Form 132 whose summary information is:
    Hypothesis Statement
    Form 165 <p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable 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: 9652, whose string of implications is:
    378 \(\Rightarrow\) 132

The conclusion Form 249 \( \not \Rightarrow \) Form 378 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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