Axiom Of Choice

This non-implication, Form 167 \( \not \Rightarrow \) 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: 9946, whose string of implications is: 379 \(\Rightarrow\) 167

A proven non-implication whose code is 5. In this case, it's Code 3: 727, Form 379 \( \not \Rightarrow \) Form 155 whose summary information is:

Hypothesis	Statement
Form 379	<p> \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). </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 167 \( \not \Rightarrow \) 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 \(A\)