Implications

This non-implication, Form 276 \( \not \Rightarrow \) Form 402, 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: 4332, whose string of implications is:
    30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 276
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1313, Form 30 \( \not \Rightarrow \) Form 324 whose summary information is:
    Hypothesis Statement
    Form 30 <p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </p>

    Conclusion Statement
    Form 324 <p> \(KW(WO,WO)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets:</strong> For every well ordered set \(M\) of well orderable sets, 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-15">Form 15</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10264, whose string of implications is:
    402 \(\Rightarrow\) 324

The conclusion Form 276 \( \not \Rightarrow \) Form 402 then follows.

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

Name Statement
\(\cal M29\) Pincus' Model II Pincus constructs a generic extension \(M[I]\) of a model \(M\) of \(ZF +\) class choice \(+ GCH\) in which \(I=\bigcup_{n\in\omega}I_n\), \(I_{-1}=2\) and \(I_{n+1}\) is a denumerable set of independent functions from \(\omega\) onto \(I_n\)

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