Implications

This non-implication, Form 308-p \( \not \Rightarrow \) Form 340, 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: 859, Form 308-p \( \not \Rightarrow \) Form 10 whose summary information is:
    Hypothesis Statement
    Form 308-p <p> If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup. </p>

    Conclusion Statement
    Form 10 <p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9349, whose string of implications is:
    340 \(\Rightarrow\) 341 \(\Rightarrow\) 10

The conclusion Form 308-p \( \not \Rightarrow \) Form 340 then follows.

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

Name Statement
\(\cal N22(p)\) Makowski/Wi\'sniewski/Mostowski Model (Where \(p\) is aprime) Let \(A=\bigcup\{A_i: i\in\omega\}\) where The \(A_i\)'s are pairwisedisjoint and each has cardinality \(p\)

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