Implications

This non-implication, Form 98 \( \not \Rightarrow \) Form 41, 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: 3102, whose string of implications is:
    202 \(\Rightarrow\) 40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 98
  • A proven non-implication whose code is 5. In this case, it's Code 3: 527, Form 202 \( \not \Rightarrow \) Form 41 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 41 <p> \(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \). </p>

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

The conclusion Form 98 \( \not \Rightarrow \) Form 41 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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