Implications

This non-implication, Form 26 \( \not \Rightarrow \) Form 29, whose code is 6, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9620, whose string of implications is:
    24 \(\Rightarrow\) 26
  • A proven non-implication whose code is 5. In this case, it's Code 3: 76, Form 24 \( \not \Rightarrow \) Form 29 whose summary information is:
    Hypothesis Statement
    Form 24 <p> \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 26 \( \not \Rightarrow \) Form 29 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\(i\in\omega, |B_i|=|C_i|=\aleph_0\)

Edit | Back