Implications

This non-implication, Form 185 \( \not \Rightarrow \) Form 409, 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: 10211, whose string of implications is:
    9 \(\Rightarrow\) 185
  • A proven non-implication whose code is 3. In this case, it's Code 3: 973, Form 9 \( \not \Rightarrow \) Form 111 whose summary information is:
    Hypothesis Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4622, whose string of implications is:
    409 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 250 \(\Rightarrow\) 111

The conclusion Form 185 \( \not \Rightarrow \) Form 409 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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