Axiom Of Choice

This non-implication, Form 34 \( \not \Rightarrow \) Form 293, 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: 6133, whose string of implications is: 91 \(\Rightarrow\) 79 \(\Rightarrow\) 94 \(\Rightarrow\) 34

A proven non-implication whose code is 5. In this case, it's Code 3: 231, Form 91 \( \not \Rightarrow \) Form 293 whose summary information is:

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

Conclusion	Statement
Form 293	<p> For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. </p>

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

The conclusion Form 34 \( \not \Rightarrow \) Form 293 then follows.

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

Name	Statement
\(\cal N28\) Blass' Permutation Model	The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\)
\(\cal N37\) A variation of Blass' model, \(\cal N28\)	Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)

Implications