Implications

This non-implication, Form 277 \( \not \Rightarrow \) Form 39, whose code is 4, 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 3. In this case, it's Code 3: 1043, Form 277 \( \not \Rightarrow \) Form 131 whose summary information is:
    Hypothesis Statement
    Form 277 <p> \(E(D,VII)\): Every non-well-orderable cardinal is decomposable. </p>

    Conclusion Statement
    Form 131 <p> \(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 783, whose string of implications is:
    39 \(\Rightarrow\) 8 \(\Rightarrow\) 126 \(\Rightarrow\) 131

The conclusion Form 277 \( \not \Rightarrow \) Form 39 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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