Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 15, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 256, Form 204 \( \not \Rightarrow \) Form 88 whose summary information is:
    Hypothesis Statement
    Form 204 <p> For every infinite \(X\), there is a function from \(X\) onto \(2X\). </p>

    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: 4296, whose string of implications is:
    15 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88

The conclusion Form 204 \( \not \Rightarrow \) Form 15 then follows.

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

Name Statement
\(\cal M2\) Feferman's model Add a denumerable number of generic reals to the base model, but do not collect them
\(\cal N9\) 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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