Implications

This non-implication, Form 355 \( \not \Rightarrow \) Form 325, 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: 1698, whose string of implications is:
    15 \(\Rightarrow\) 322 \(\Rightarrow\) 355
  • A proven non-implication whose code is 3. In this case, it's Code 3: 4, Form 15 \( \not \Rightarrow \) Form 17 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 17 <p> <strong>Ramsey's Theorem I:</strong> If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see <a href="/form-classes/howard-rubin-325">Form 325</a>.), <a href="/books/8">Jech [1973b]</a>, p 164 prob 11.20. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9902, whose string of implications is:
    325 \(\Rightarrow\) 17

The conclusion Form 355 \( \not \Rightarrow \) Form 325 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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