Implications

This non-implication, Form 366 \( \not \Rightarrow \) Form 103, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6202, whose string of implications is:
    91 \(\Rightarrow\) 79 \(\Rightarrow\) 367 \(\Rightarrow\) 366
  • A proven non-implication whose code is 5. In this case, it's Code 3: 198, Form 91 \( \not \Rightarrow \) Form 103 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

    Conclusion Statement
    Form 103 <p> If \((P,<)\) is a linear ordering and \(|P| > \aleph_{1}\) then some initial segment of \(P\) is uncountable. <a href="/books/8">Jech [1973b]</a>, p 164 prob 11.21. </p>

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

The conclusion Form 366 \( \not \Rightarrow \) Form 103 then follows.

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

Name Statement
\(\cal N14\) Morris/Jech Model \(A = \bigcup\{A_{\alpha}: \alpha <\omega_1\}\), where the \(A_{\alpha}\)'s are pairwise disjoint, each iscountably infinite, and each is ordered like the rationals; \(\cal G\) isthe group of all permutations on \(A\) that leave each \(A_{\alpha}\) fixedand preserve the ordering on each \(A_{\alpha}\); and \(S = \{B_{\gamma}:\gamma < \omega_1\}\), where \(B_{\gamma}= \bigcup\{A_{\alpha}: \alpha <\gamma\}\)

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