Implications

This non-implication, Form 350 \( \not \Rightarrow \) Form 375, 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: 1891, whose string of implications is:
    23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 350
  • A proven non-implication whose code is 5. In this case, it's Code 3: 56, Form 23 \( \not \Rightarrow \) Form 155 whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5922, whose string of implications is:
    375 \(\Rightarrow\) 78 \(\Rightarrow\) 155

The conclusion Form 350 \( \not \Rightarrow \) Form 375 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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