Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 20, whose code is 4, 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: 9941, whose string of implications is:
    40 \(\Rightarrow\) 165
  • A proven non-implication whose code is 3. In this case, it's Code 3: 102, Form 40 \( \not \Rightarrow \) Form 347 whose summary information is:
    Hypothesis Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </p>

    Conclusion Statement
    Form 347 <p> <strong>Idemmultiple Partition Principle</strong>: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1813, whose string of implications is:
    20 \(\Rightarrow\) 101 \(\Rightarrow\) 347

The conclusion Form 165 \( \not \Rightarrow \) Form 20 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
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\) Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above

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