Implications

This non-implication, Form 93 \( \not \Rightarrow \) Form 100, 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: 4333, whose string of implications is:
    30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 93
  • 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: 7092, whose string of implications is:
    100 \(\Rightarrow\) 347 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 16 \(\Rightarrow\) 6

The conclusion Form 93 \( \not \Rightarrow \) Form 100 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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