Implications

This non-implication, Form 186 \( \not \Rightarrow \) Form 67, whose code is 6, 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: 9771, whose string of implications is:
    187 \(\Rightarrow\) 186
  • A proven non-implication whose code is 5. In this case, it's Code 3: 454, Form 187 \( \not \Rightarrow \) Form 67 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 67 <p> \(MC(\infty,\infty)\) \((MC)\), <strong>The Axiom of Multiple Choice:</strong> For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). </p>

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

The conclusion Form 186 \( \not \Rightarrow \) Form 67 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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