Implications

This non-implication, Form 403 \( \not \Rightarrow \) Form 100, 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: 7867, whose string of implications is:
    202 \(\Rightarrow\) 398 \(\Rightarrow\) 400 \(\Rightarrow\) 403
  • A proven non-implication whose code is 5. In this case, it's Code 3: 531, Form 202 \( \not \Rightarrow \) Form 100 whose summary information is:
    Hypothesis Statement
    Form 202 <p> \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function. </p>

    Conclusion Statement
    Form 100 <p> <strong>Weak Partition Principle:</strong>  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\). </p>

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

The conclusion Form 403 \( \not \Rightarrow \) Form 100 then follows.

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

Name Statement
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false

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