Axiom Of Choice

This non-implication, Form 293 $ \not \Rightarrow $ Form 15, whose code is 4, 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: 2010, whose string of implications is: 295 $\Rightarrow$ 30 $\Rightarrow$ 293

A proven non-implication whose code is 3. In this case, it's Code 3: 756, Form 295 $ \not \Rightarrow $ Form 15 whose summary information is:

Hypothesis	Statement
Form 295	<p> <strong>DO:</strong> Every infinite set has a dense linear ordering. </p>

Conclusion	Statement
Form 15	<p> $KW(\infty,\infty)$ (KW), <strong>The Kinna-Wagner Selection Principle:</strong> For every set $M$ there is a function $f$ such that for all $A\in M$, if $\|A\|>1$ then $\emptyset\neq f(A)\subsetneq A$. (See <a href="/form-classes/howard-rubin-81($n$)">Form 81($n$)</a>. </p>

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

The conclusion Form 293 $ \not \Rightarrow $ Form 15 then follows.

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

Name	Statement
$\cal M29$ Pincus' Model II	Pincus constructs a generic extension $M[I]$ of a model $M$ of $ZF +$ class choice $+ GCH$ in which $I=\bigcup_{n\in\omega}I_n$, $I_{-1}=2$ and $I_{n+1}$ is a denumerable set of independent functions from $\omega$ onto $I_n$
$\cal M44$ Pincus' Model VI	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 $(B)$
$\cal N38$ Howard/Rubin Model I	Let $(A,\le)$ be an ordered set of atomswhich is order isomorphic to ${\Bbb Q}^\omega$, the set of all functionsfrom $\omega$ into $\Bbb Q$ ordered by the lexicographic ordering
$\cal N40$ Howard/Rubin Model II	A variation of $\cal N38$
$\cal N48$ Pincus' Model XI	$\cal A=(A,<,C_0,C_1,\dots)$ is called an<em>ordered colored set</em> (OC set) if $<$ is a linear ordering on $A$and the $C_i$, for $i\in\omega$ are subsets of $A$ such that for each$a\in A$ there is exactly one $n\in\omega$ such that $a\in C_n$

Name	Statement
\(\cal M29\) Pincus' Model II	Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)
\(\cal M44\) Pincus' Model VI	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((B)\)
\(\cal N38\) Howard/Rubin Model I	Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering
\(\cal N40\) Howard/Rubin Model II	A variation of \(\cal N38\)
\(\cal N48\) Pincus' Model XI	\(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\)