Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 96, 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: 140, Form 0 \( \not \Rightarrow \) Form 235 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 235 <p> If \(V\) is a vector space and \(B_{1}\) and \(B_{2}\) are bases for \(V\) then \(|B_{1}|\) and \(|B_{2}|\) are comparable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10312, whose string of implications is:
    96 \(\Rightarrow\) 235

The conclusion Form 0 \( \not \Rightarrow \) Form 96 then follows.

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

Name Statement
\(\cal N13\) L\"auchli/Jech Model \(A = B_1\cup B_2\), where \(B_1=\bigcup\{A_{j1} : j\in\omega\}\), and \(B_2 = \bigcup\{A_{j2} :j\in\omega\}\), and each \(A_{ji}\) is a 6-element set

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