Implications

This non-implication, Form 329 \( \not \Rightarrow \) Form 21, whose code is 4, 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: 9974, whose string of implications is:
    60 \(\Rightarrow\) 329
  • A proven non-implication whose code is 3. In this case, it's Code 3: 73, Form 60 \( \not \Rightarrow \) Form 31 whose summary information is:
    Hypothesis Statement
    Form 60 <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </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: 1849, whose string of implications is:
    21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31

The conclusion Form 329 \( \not \Rightarrow \) Form 21 then follows.

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

Name Statement
\(\cal M20\) 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 \(\aleph_1\)

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