Implications

This non-implication, Form 32 \( \not \Rightarrow \) Form 359, 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: 111, Form 32 \( \not \Rightarrow \) Form 31 whose summary information is:
    Hypothesis Statement
    Form 32 <p> \(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function. </p>

    Conclusion Statement
    Form 31 <p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong>  The union of a denumerable set of denumerable sets is denumerable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1760, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31

The conclusion Form 32 \( \not \Rightarrow \) Form 359 then follows.

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

Name Statement
\(\cal M12(\aleph)\) Truss' Model I This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular
\(\cal M20\) Felgner's Model I Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\)
\(\cal N18\) Howard's Model I Let \(B= {B_n: n\in\omega}\) where the \(B_n\)'sare pairwise disjoint and each is countably infinite and let \(A=\bigcup B\)

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