Implications

This non-implication, Form 13 \( \not \Rightarrow \) Form 379, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6935, whose string of implications is:
    337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 13
  • A proven non-implication whose code is 5. In this case, it's Code 3: 699, Form 337 \( \not \Rightarrow \) Form 342-n whose summary information is:
    Hypothesis Statement
    Form 337 <p> \(C(WO\), <strong>uniformly linearly ordered</strong>):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\). </p>

    Conclusion Statement
    Form 342-n <p> (For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See <a href="/form-classes/howard-rubin-166">Form 166</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5905, whose string of implications is:
    379 \(\Rightarrow\) 73 \(\Rightarrow\) 342-n

The conclusion Form 13 \( \not \Rightarrow \) Form 379 then follows.

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

Name Statement
\(\cal N50(E)\) Brunner's Model III \(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)

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