Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 236, whose code is 6, 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: 11016, whose string of implications is:
    191 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 497, Form 191 \( \not \Rightarrow \) Form 236 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 236 <p> If \(V\) is a vector space with a basis and \(S\) is a linearly independent subset of \(V\) such that no proper extension of \(S\) is a basis for \(V\), then \(S\) is a basis for \(V\). </p>

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

The conclusion Form 0 \( \not \Rightarrow \) Form 236 then follows.

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

Name Statement
\(\cal N44\) Gross' model \(A\) is a vector space over a finite field withbasis \(B = \bigcup_{i\in \omega} B_i\) where the \(B_i\) are pairwisedisjoint and \(|B_i| = 4\) for each \(i\in\omega\)

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