Implications

This non-implication, Form 26 \( \not \Rightarrow \) Form 155, 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: 9620, whose string of implications is:
    24 \(\Rightarrow\) 26
  • A proven non-implication whose code is 5. In this case, it's Code 3: 89, Form 24 \( \not \Rightarrow \) Form 155 whose summary information is:
    Hypothesis Statement
    Form 24 <p> \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function. </p>

    Conclusion Statement
    Form 155  \(LC\): There are no non-trivial L&auml;uchli continua. (A <em>L&auml;uchli continuum</em> is a strongly connected continuum. <em>Continuum</em> \(\equiv\) compact, connected, Hausdorff space; and <em>strongly connected</em> \(\equiv\) every continuous real valued function is constant.) </p>

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

The conclusion Form 26 \( \not \Rightarrow \) Form 155 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

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