Implications

This non-implication, Form 4 \( \not \Rightarrow \) Form 284, whose code is 6, 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: 9709, whose string of implications is:
    3 \(\Rightarrow\) 4
  • A proven non-implication whose code is 5. In this case, it's Code 3: 3, Form 3 \( \not \Rightarrow \) Form 88 whose summary information is:
    Hypothesis Statement
    Form 3  \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

    Conclusion Statement
    Form 88 <p>  \(C(\infty ,2)\):  Every family of pairs has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4294, whose string of implications is:
    284 \(\Rightarrow\) 61 \(\Rightarrow\) 88

The conclusion Form 4 \( \not \Rightarrow \) Form 284 then follows.

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

Name Statement
\(\cal N9\) Halpern/Howard Model \(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\)

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