Implications

This non-implication, Form 92 \( \not \Rightarrow \) Form 359, 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: 3513, whose string of implications is:
    40 \(\Rightarrow\) 337 \(\Rightarrow\) 92
  • 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: 1819, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 101 \(\Rightarrow\) 347

The conclusion Form 92 \( \not \Rightarrow \) 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
\(\cal N12(\aleph_2)\) Another variation of \(\cal N1\) Change "\(\aleph_1\)" to "\(\aleph_2\)" in \(\cal N12(\aleph_1)\) above

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