Implications

This non-implication, Form 273 \( \not \Rightarrow \) Form 191, 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: 9365, whose string of implications is:
    363 \(\Rightarrow\) 364 \(\Rightarrow\) 273
  • A proven non-implication whose code is 5. In this case, it's Code 3: 717, Form 363 \( \not \Rightarrow \) Form 190 whose summary information is:
    Hypothesis Statement
    Form 363 <p>There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). <a href="/books/Moore-1982">G. Moore [1982]</a>, p 325. </p>

    Conclusion Statement
    Form 190 <p> There is a non-trivial injective Abelian group. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7695, whose string of implications is:
    191 \(\Rightarrow\) 189 \(\Rightarrow\) 190

The conclusion Form 273 \( \not \Rightarrow \) Form 191 then follows.

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

Name Statement
\(\cal N28\) Blass' Permutation Model The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\)

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