Axiom Of Choice

This non-implication, Form 327 \( \not \Rightarrow \) Form 144, 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: 7504, whose string of implications is: 165 \(\Rightarrow\) 122 \(\Rightarrow\) 327
A proven non-implication whose code is 3. In this case, it's Code 3: 1058, Form 165 \( \not \Rightarrow \) Form 144 whose summary information is:

Hypothesis Statement

Form 165 <p> \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. </p>

Conclusion Statement

Form 144 <p> Every set is almost well orderable. </p>
This non-implication was constructed without the use of this last code 2/1 implication

Hypothesis	Statement
Form 165	<p> \(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function. </p>

Conclusion	Statement
Form 144	<p> Every set is almost well orderable. </p>

The conclusion Form 327 \( \not \Rightarrow \) Form 144 then follows.

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

Name	Statement
\(\cal M1\) Cohen's original model	Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them
\(\cal M3\) Mathias' model	Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)
\(\cal M11\) Forti/Honsell Model	Using a model of \(ZF + V = L\) for the ground model, the authors construct a generic extension, \(\cal M\), using Easton forcing which adds \(\kappa\) generic subsets to each regular cardinal \(\kappa\)
\(\cal M20\) Felgner's Model I	Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\)
\(\cal M40(\kappa)\) Pincus' Model IV	The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N3\) Mostowski's Linearly Ordered Model	\(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)
\(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\))	\(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\)
\(\cal N19(\precsim)\) 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 \(I=P\cup Q\)
\(\cal N29\) Dawson/Howard Model	Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\)
\(\cal N38\) 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 \(a\in C_n\)

Implications