Implications

This non-implication, Form 47-n \( \not \Rightarrow \) Form 1, 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: 283, Form 47-n \( \not \Rightarrow \) Form 422-n whose summary information is:
    Hypothesis Statement
    Form 47-n <p> If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. </p>

    Conclusion Statement
    Form 422-n <p> \(UT(WO,n,WO)\), \(n\in \omega-\{0,1\}\): The union of a well ordered set of \(n\) element sets can be well ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10733, whose string of implications is:
    1 \(\Rightarrow\) 422-n

The conclusion Form 47-n \( \not \Rightarrow \) Form 1 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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