Axiom Of Choice

This non-implication, Form 361 \( \not \Rightarrow \) Form 359, 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: 10135, whose string of implications is: 8 \(\Rightarrow\) 361

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>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1751, whose string of implications is: 359 \(\Rightarrow\) 20 \(\Rightarrow\) 21 \(\Rightarrow\) 23 \(\Rightarrow\) 151

The conclusion Form 361 \( \not \Rightarrow \) Form 359 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