Implications

This non-implication, Form 30 \( \not \Rightarrow \) Form 421, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 236, Form 30 \( \not \Rightarrow \) Form 5 whose summary information is:
    Hypothesis Statement
    Form 30 <p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </p>

    Conclusion Statement
    Form 5 <p> \(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9331, whose string of implications is:
    421 \(\Rightarrow\) 338 \(\Rightarrow\) 32 \(\Rightarrow\) 5

The conclusion Form 30 \( \not \Rightarrow \) Form 421 then follows.

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

Name Statement
\(\cal M6\) Sageev's Model I Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)

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