Implications

This non-implication, Form 374-n \( \not \Rightarrow \) Form 16, 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: 1245, whose string of implications is:
    30 \(\Rightarrow\) 10 \(\Rightarrow\) 423 \(\Rightarrow\) 374-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 265, Form 30 \( \not \Rightarrow \) Form 6 whose summary information is:
    Hypothesis Statement
    Form 30 <p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </p>

    Conclusion Statement
    Form 6 <p> \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9876, whose string of implications is:
    16 \(\Rightarrow\) 6

The conclusion Form 374-n \( \not \Rightarrow \) Form 16 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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