Implications

This non-implication, Form 420 \( \not \Rightarrow \) Form 111, 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: 2356, whose string of implications is:
    31 \(\Rightarrow\) 419 \(\Rightarrow\) 420
  • A proven non-implication whose code is 3. In this case, it's Code 3: 974, Form 31 \( \not \Rightarrow \) Form 111 whose summary information is:
    Hypothesis 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>

    Conclusion Statement
    Form 111 <p> \(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. </p>

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

The conclusion Form 420 \( \not \Rightarrow \) Form 111 then follows.

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

Name Statement
\(\cal N2(\aleph_{\alpha})\) Jech's Model This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)

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