Axiom Of Choice

This non-implication, Form 13 \( \not \Rightarrow \) Form 147, 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: 940, whose string of implications is: 214 \(\Rightarrow\) 9 \(\Rightarrow\) 13

A proven non-implication whose code is 3. In this case, it's Code 3: 941, Form 214 \( \not \Rightarrow \) Form 91 whose summary information is:

Hypothesis	Statement
Form 214	<p> \(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(\|f(y)\|=\aleph_{0}\). </p>

Conclusion	Statement
Form 91	<p> \(PW\): The power set of a well ordered set can be well ordered. </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9630, whose string of implications is: 147 \(\Rightarrow\) 91

The conclusion Form 13 \( \not \Rightarrow \) Form 147 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 \(\{I_{\alpha}:\alpha \le \lambda\}\)
\(\cal M29\) 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 \(I_n\)
\(\cal M43\) 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 \((E)\)

Implications