Axiom Of Choice

This non-implication, Form 35 \( \not \Rightarrow \) Form 198, 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: 6934, whose string of implications is: 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 35

A proven non-implication whose code is 5. In this case, it's Code 3: 693, Form 337 \( \not \Rightarrow \) Form 198 whose summary information is:

Hypothesis	Statement
Form 337	<p> \(C(WO\), <strong>uniformly linearly ordered</strong>): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </p>

Conclusion	Statement
Form 198	<p> For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset. </p>

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

The conclusion Form 35 \( \not \Rightarrow \) Form 198 then follows.

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

Name	Statement
\(\cal N2\) The Second Fraenkel Model	The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).

Implications