Implications

This non-implication, Form 357 \( \not \Rightarrow \) Form 192, 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: 8939, whose string of implications is:
    322 \(\Rightarrow\) 324 \(\Rightarrow\) 357
  • A proven non-implication whose code is 5. In this case, it's Code 3: 646, Form 322 \( \not \Rightarrow \) Form 154 whose summary information is:
    Hypothesis Statement
    Form 322 <p> \(KW(WO,\infty)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of sets:</strong> For every  well ordered 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-15">Form 15</a>). </p>

    Conclusion Statement
    Form 154 <p> <strong>Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces:</strong> The product of countably many \(T_2\) compact spaces is compact. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3879, whose string of implications is:
    192 \(\Rightarrow\) 43 \(\Rightarrow\) 154

The conclusion Form 357 \( \not \Rightarrow \) Form 192 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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