Axiom Of Choice

This non-implication, Form 90 \( \not \Rightarrow \) Form 114, whose code is 6, is constructed around a proven non-implication as follows:

This non-implication was constructed without the use of this first code 2/1 implication.

A proven non-implication whose code is 5. In this case, it's Code 3: 189, Form 90 \( \not \Rightarrow \) Form 114 whose summary information is:

Hypothesis	Statement
Form 90	<p> \(LW\): Every linearly ordered set can be well ordered. <a href="/books/8">Jech [1973b]</a>, p 133. </p>

Conclusion	Statement
Form 114	<p> Every A-bounded \(T_2\) topological space is weakly Loeb. (<em>\(A\)-bounded</em> means amorphous subsets are relatively compact. <em>Weakly Loeb</em> means the set of non-empty closed subsets has a multiple choice function.) </p>

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

The conclusion Form 90 \( \not \Rightarrow \) Form 114 then follows.

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

Name	Statement
\(\cal N4\) The Mathias/Pincus Model I	\(A\) is countably infinite;\(\precsim\) is a universal homogeneous partial ordering on \(A\) (See<a href="/articles/Jech-1973b">Jech [1973b]</a> p 101 for definitions.); \(\cal G\) is the group ofall order automorphisms on \((A,\precsim)\); and \(S\) is the set of allfinite subsets of \(A\)