Axiom Of Choice

This non-implication, Form 24 \( \not \Rightarrow \) Form 59-le, 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: 78, Form 24 \( \not \Rightarrow \) Form 59-le whose summary information is:

Hypothesis	Statement
Form 24	<p> \(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function. </p>

Conclusion	Statement
Form 59-le	<p> If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\).<br /> <a href="/articles/Mathias-1979">Mathias [1979]</a>, p 120. </p>

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

The conclusion Form 24 \( \not \Rightarrow \) Form 59-le then follows.

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

Name	Statement
\(\cal N19(\precsim)\) Tsukada's Model	Let \((P,\precsim)\) be a partiallyordered set that is not well ordered; Let \(Q\) be a countably infinite set,disjoint from \(P\); and let \(I=P\cup Q\)