Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 50, 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: 10761, whose string of implications is:
    17 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 32, Form 17 \( \not \Rightarrow \) Form 69 whose summary information is:
    Hypothesis 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>

    Conclusion Statement
    Form 69 <p> Every field has an algebraic closure.  <a href="/books/8">Jech [1973b]</a>, p 13. <p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1361, whose string of implications is:
    50 \(\Rightarrow\) 14 \(\Rightarrow\) 69

The conclusion Form 0 \( \not \Rightarrow \) Form 50 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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