Axiom Of Choice

This non-implication, Form 382 \( \not \Rightarrow \) Form 375, 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: 10026, whose string of implications is: 381 \(\Rightarrow\) 382

A proven non-implication whose code is 5. In this case, it's Code 3: 755, Form 381 \( \not \Rightarrow \) Form 155 whose summary information is:

Hypothesis	Statement
Form 381	<p> <strong>DUM</strong>: The disjoint union of metrizable spaces is metrizable. </p>

Conclusion	Statement
Form 155	\(LC\): There are no non-trivial Läuchli continua. (A <em>Läuchli 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 382 \( \not \Rightarrow \) 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 \(A\)