Implications

This non-implication, Form 316 \( \not \Rightarrow \) Form 168, 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: 6515, whose string of implications is:
    112 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 316
  • A proven non-implication whose code is 5. In this case, it's Code 3: 260, Form 112 \( \not \Rightarrow \) Form 126 whose summary information is:
    Hypothesis Statement
    Form 112 <p> \(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). </p>

    Conclusion Statement
    Form 126 <p> \(MC(\aleph_0,\infty)\), <strong>Countable axiom of multiple choice:</strong> For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7193, whose string of implications is:
    168 \(\Rightarrow\) 100 \(\Rightarrow\) 347 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 126

The conclusion Form 316 \( \not \Rightarrow \) Form 168 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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