Implications

This non-implication, Form 39 \( \not \Rightarrow \) Form 359, 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: 3264, whose string of implications is:
    202 \(\Rightarrow\) 40 \(\Rightarrow\) 39
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1820, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 101 \(\Rightarrow\) 100

The conclusion Form 39 \( \not \Rightarrow \) Form 359 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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