Implications

This non-implication, Form 53 \( \not \Rightarrow \) Form 399, 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: 10161, whose string of implications is:
    3 \(\Rightarrow\) 53
  • 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: 8992, whose string of implications is:
    399 \(\Rightarrow\) 323 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88

The conclusion Form 53 \( \not \Rightarrow \) Form 399 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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