Axiom Of Choice

This non-implication, Form 111 \( \not \Rightarrow \) Form 149, 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: 2258, whose string of implications is: 295 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 121 \(\Rightarrow\) 122 \(\Rightarrow\) 250 \(\Rightarrow\) 111
A proven non-implication whose code is 3. In this case, it's Code 3: 980, Form 295 \( \not \Rightarrow \) Form 118 whose summary information is:

Hypothesis Statement

Form 295 DO: Every infinite set has a dense linear ordering. 

Conclusion Statement

Form 118 Every linearly orderable topological space is normal. <a href="/books/28">Birkhoff [1967]</a>, p 241.
An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5345, whose string of implications is: 149 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 118

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

Conclusion	Statement
Form 118	<p> Every linearly orderable topological space is normal. <a href="/books/28">Birkhoff [1967]</a>, p 241. </p>

The conclusion Form 111 \( \not \Rightarrow \) Form 149 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 \(a\in C_n\)