Axiom Of Choice

This non-implication, Form 99 \( \not \Rightarrow \) Form 98, 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: 246, Form 99 \( \not \Rightarrow \) Form 98 whose summary information is:

Hypothesis	Statement
Form 99	<p> <strong>Rado's Selection Lemma:</strong> Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. </p>

Conclusion	Statement
Form 98	<p> The set of all finite subsets of a Dedekind finite set is Dedekind finite. <a href="/books/8">Jech [1973b]</a> p 161 prob 11.5. </p>

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

The conclusion Form 99 \( \not \Rightarrow \) Form 98 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 \( A \).