Axiom Of Choice

This non-implication, Form 92 \( \not \Rightarrow \) Form 163, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10235, whose string of implications is: 337 \(\Rightarrow\) 92

A proven non-implication whose code is 5. In this case, it's Code 3: 690, Form 337 \( \not \Rightarrow \) Form 163 whose summary information is:

Hypothesis	Statement
Form 337	<p> \(C(WO\), <strong>uniformly linearly ordered</strong>): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </p>

Conclusion	Statement
Form 163	<p> Every non-well-orderable set has an infinite, Dedekind finite subset. </p>

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

The conclusion Form 92 \( \not \Rightarrow \) Form 163 then follows.

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

Name	Statement
\(\cal M2\) Feferman's model	Add a denumerable number of generic reals to the base model, but do not collect them
\(\cal M13\) Feferman/Solovay Model	This model is an extension of <a href="/models/Feferman-1">\(\cal M2\)</a> in which there are \(\omega_1\) generic real numbers, but no set to collect them
\(\cal M25\) Freyd's Model	Using topos-theoretic methods due to Fourman, Freyd constructs a Boolean-valued model of \(ZF\) in which every well ordered family of sets has a choice function (<a href="/form-classes/howard-rubin-40">Form 40</a> is true), but \(C(\|\Bbb R\|,\infty)\) (<a href="/form-classes/howard-rubin-181">Form 181</a>) is false
\(\cal N2\) 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 \( A \).
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\)	Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\)	Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above
\(\cal N33\) Howard/H\.Rubin/J\.Rubin 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 boundedsubsets of \(A\)

Implications