Implications

This non-implication, Form 84 \( \not \Rightarrow \) Form 105, 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: 1684, whose string of implications is:
    15 \(\Rightarrow\) 82 \(\Rightarrow\) 84
  • A proven non-implication whose code is 3. In this case, it's Code 3: 77, Form 15 \( \not \Rightarrow \) Form 105 whose summary information is:
    Hypothesis Statement
    Form 15 <p> \(KW(\infty,\infty)\) (KW), <strong>The Kinna-Wagner Selection Principle:</strong> For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See <a href="/form-classes/howard-rubin-81($n$)">Form 81(\(n\))</a>.   </p>

    Conclusion Statement
    Form 105 <p> There is a  partially ordered set \((A,\le)\) such that for no set \(B\) is \((B,\le)\) (the ordering  on \(B\) is the usual injective cardinal ordering) isomorphic to \((A,\le)\). </p>

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

The conclusion Form 84 \( \not \Rightarrow \) Form 105 then follows.

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

Name Statement
\(\cal M11\) Forti/Honsell Model Using a model of \(ZF + V = L\) for the ground model, the authors construct a generic extension, \(\cal M\), using Easton forcing which adds \(\kappa\) generic subsets to each regular cardinal \(\kappa\)

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