Implications

This non-implication, Form 355 \( \not \Rightarrow \) Form 177, whose code is 6, 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: 10005, whose string of implications is:
    322 \(\Rightarrow\) 355
  • A proven non-implication whose code is 5. In this case, it's Code 3: 647, Form 322 \( \not \Rightarrow \) Form 177 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 177 <p> An infinite box product of regular \(T_1\) spaces, each of cardinality greater than 1, is neither first countable nor connected. </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 355 \( \not \Rightarrow \) Form 177 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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