Axiom Of Choice

This non-implication, Form 190 \( \not \Rightarrow \) Form 216, 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: 7695, whose string of implications is: 191 \(\Rightarrow\) 189 \(\Rightarrow\) 190

A proven non-implication whose code is 5. In this case, it's Code 3: 492, Form 191 \( \not \Rightarrow \) Form 216 whose summary information is:

Hypothesis	Statement
Form 191	<p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

Conclusion	Statement
Form 216	<p> Every infinite tree has either an infinite chain or an infinite antichain. </p>

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

The conclusion Form 190 \( \not \Rightarrow \) Form 216 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 \( A \).