Implications

This non-implication, Form 31 \( \not \Rightarrow \) Form 20, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1708, whose string of implications is:
    16 \(\Rightarrow\) 352 \(\Rightarrow\) 31
  • A proven non-implication whose code is 5. In this case, it's Code 3: 7, Form 16 \( \not \Rightarrow \) Form 29 whose summary information is:
    Hypothesis Statement
    Form 16 <p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function. </p>

    Conclusion Statement
    Form 29  If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup^{}_{x\in S} A_{x}| = |\bigcup^{}_{x\in S} B_{x}|\). <a href="/books/2">Moore, G. [1982]</a>, p 324. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10096, whose string of implications is:
    20 \(\Rightarrow\) 29

The conclusion Form 31 \( \not \Rightarrow \) Form 20 then follows.

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

Name Statement
\(\cal N39\) Howard's Model II \(A\) is denumerable and is a disjoint union\(\bigcup_{i\in\omega}B_i\cup\bigcup_{i\in\omega}C_i\), where for all\(i\in\omega, |B_i|=|C_i|=\aleph_0\)

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