Implications

This non-implication, Form 222 \( \not \Rightarrow \) Form 292, 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: 4856, whose string of implications is:
    133 \(\Rightarrow\) 63 \(\Rightarrow\) 70 \(\Rightarrow\) 222
  • A proven non-implication whose code is 5. In this case, it's Code 3: 355, Form 133 \( \not \Rightarrow \) Form 292 whose summary information is:
    Hypothesis Statement
    Form 133  <p> Every set is either well orderable or has an infinite amorphous subset. </p>

    Conclusion Statement
    Form 292 <p> \(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\). </p>

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

The conclusion Form 222 \( \not \Rightarrow \) Form 292 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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