Implications

This non-implication, Form 198 \( \not \Rightarrow \) Form 296, 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: 2012, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 198
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1256, Form 295 \( \not \Rightarrow \) Form 296 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </p>

    Conclusion Statement
    Form 296 <p> <strong>Part-\(\infty\):</strong> Every infinite set is the disjoint union of infinitely many infinite sets. </p>

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

The conclusion Form 198 \( \not \Rightarrow \) Form 296 then follows.

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

Name Statement
\(\cal N48\) Pincus' Model XI \(\cal A=(A,<,C_0,C_1,\dots)\) is called an<em>ordered colored set</em> (OC set) if \(<\) is a linear ordering on \(A\)and the \(C_i\), for \(i\in\omega\) are subsets of \(A\) such that for each\(a\in A\) there is exactly one \(n\in\omega\) such that \(a\in C_n\)

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