Implications

This non-implication, Form 295 \( \not \Rightarrow \) Form 400, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1321, Form 295 \( \not \Rightarrow \) Form 324 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </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: 9556, whose string of implications is:
    400 \(\Rightarrow\) 402 \(\Rightarrow\) 324

The conclusion Form 295 \( \not \Rightarrow \) Form 400 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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