Implications

This non-implication, Form 187 \( \not \Rightarrow \) Form 239, 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: 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7880, whose string of implications is:
    239 \(\Rightarrow\) 427 \(\Rightarrow\) 67

The conclusion Form 187 \( \not \Rightarrow \) Form 239 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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