Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 242, 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: 501, Form 191 \( \not \Rightarrow \) Form 241 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 241 <p> Every algebraic closure of \(\Bbb Q\) has a real closed subfield. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10230, whose string of implications is:
    242 \(\Rightarrow\) 241

The conclusion Form 0 \( \not \Rightarrow \) Form 242 then follows.

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

Name Statement
\(\cal N31\) L&auml;uchli's Model IV The set \(A\) is denumerable

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