Implications

This non-implication, Form 204 \( \not \Rightarrow \) Form 408, 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: 4680, whose string of implications is:
    408 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88

The conclusion Form 204 \( \not \Rightarrow \) Form 408 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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