Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 1, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10746, whose string of implications is:
    6 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 5, Form 6 \( \not \Rightarrow \) Form 190 whose summary information is:
    Hypothesis Statement
    Form 6 <p> \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable. </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: 10503, whose string of implications is:
    1 \(\Rightarrow\) 190

The conclusion Form 0 \( \not \Rightarrow \) Form 1 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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