Implications

This non-implication, Form 190 \( \not \Rightarrow \) Form 103, whose code is 6, 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: 7695, whose string of implications is:
    191 \(\Rightarrow\) 189 \(\Rightarrow\) 190
  • A proven non-implication whose code is 5. In this case, it's Code 3: 473, Form 191 \( \not \Rightarrow \) Form 103 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </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 190 \( \not \Rightarrow \) 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 \(B_{\gamma}= \bigcup\{A_{\alpha}: \alpha <\gamma\}\)

Edit | Back