Implications

This non-implication, Form 84 \( \not \Rightarrow \) Form 395, 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: 1335, Form 15 \( \not \Rightarrow \) Form 330 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 330 <p> \(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See <a href="/form-classes/howard-rubin-67">Form 67</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9536, whose string of implications is:
    395 \(\Rightarrow\) 396 \(\Rightarrow\) 330

The conclusion Form 84 \( \not \Rightarrow \) Form 395 then follows.

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

Name Statement
\(\cal M34(\aleph_1)\) Pincus' Model III Pincus proves that Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a> can be extended by adding \(\aleph_1\) generic sets along with the set \(b\) containing them and well orderings of all countable subsets of \(b\)

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