Axiom Of Choice

This non-implication, Form 35 \( \not \Rightarrow \) Form 151, 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: 671, whose string of implications is: 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 35

A proven non-implication whose code is 3. In this case, it's Code 3: 53, Form 8 \( \not \Rightarrow \) Form 151 whose summary information is:

Hypothesis	Statement
Form 8	<p> \(C(\aleph_{0},\infty)\): </p>

Conclusion	Statement
Form 151	<p> \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see <a href="/notes/note-27">note 27</a> for \(UT(WO,\kappa,WO)\).) </p>

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

The conclusion Form 35 \( \not \Rightarrow \) Form 151 then follows.

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

Name	Statement
\(\cal M1(\langle\omega_1\rangle)\) Cohen/Pincus Model	Pincus extends the methods of Cohen and adds a generic \(\omega_1\)-sequence, \(\langle I_{\alpha}: \alpha\in\omega_1\rangle\), of denumerable sets, where \(I_0\) is a denumerable set of generic reals, each \(I_{\alpha+1}\) is a generic set of enumerations of \(I_{\alpha}\), and for a limit ordinal \(\lambda\),\(I_{\lambda}\) is a generic set of choice functions for \(\{I_{\alpha}:\alpha \le \lambda\}\)
\(\cal M43\) Pincus' Model V	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((A)\)
\(\cal M44\) Pincus' Model VI	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((B)\)
\(\cal M45\) Pincus' Model VII	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\)
\(\cal M46(m,M)\) Pincus' Model VIII	This model depends on the natural number \(m\) and the set of natural numbers \(M\) which must satisfy Mostowski's condition: <ul type="none"> <li>\(S(M,m)\): For everydecomposition \(m = p_{1} + \ldots + p_{s}\) of \(m\) into a sum of primes at least one \(p_{i}\) divides an element of \(M\)</li> </ul>
\(\cal M47(n,M)\) Pincus' Model IX	This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)
\(\cal N2(\aleph_{\alpha})\) Jech's Model	This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)
\(\cal N15\) Brunner/Howard Model I	\(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\)

Implications