Implications

This non-implication, Form 178-n-N \( \not \Rightarrow \) Form 329, 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: 1681, whose string of implications is:
    15 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 178-n-N
  • 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: 10266, whose string of implications is:
    329 \(\Rightarrow\) 330

The conclusion Form 178-n-N \( \not \Rightarrow \) Form 329 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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