Implications

This non-implication, Form 84 \( \not \Rightarrow \) Form 49, 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: 127, Form 15 \( \not \Rightarrow \) Form 49 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 49 <p> <strong>Order Extension Principle:</strong> Every partial ordering can be extended to a linear ordering.  <a href="/articles/Tarski-1924">Tarski [1924]</a>, p 78. </p>

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

The conclusion Form 84 \( \not \Rightarrow \) Form 49 then follows.

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

Name Statement
\(\cal M3\) Mathias' model Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)

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