Implications

This non-implication, Form 187 \( \not \Rightarrow \) Form 62, whose code is 6, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 453, Form 187 \( \not \Rightarrow \) Form 45-n whose summary information is:
    Hypothesis Statement
    Form 187 <p> Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.) </p>

    Conclusion Statement
    Form 45-n <p> If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4265, whose string of implications is:
    62 \(\Rightarrow\) 61 \(\Rightarrow\) 45-n

The conclusion Form 187 \( \not \Rightarrow \) Form 62 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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