Implications

This non-implication, Form 16 \( \not \Rightarrow \) Form 130, whose code is 4, 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: 401, whose string of implications is:
    181 \(\Rightarrow\) 8 \(\Rightarrow\) 16
  • A proven non-implication whose code is 3. In this case, it's Code 3: 254, Form 181 \( \not \Rightarrow \) Form 203 whose summary information is:
    Hypothesis Statement
    Form 181 <p> \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function. </p>

    Conclusion Statement
    Form 203 <p> \(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5997, whose string of implications is:
    130 \(\Rightarrow\) 79 \(\Rightarrow\) 203

The conclusion Form 16 \( \not \Rightarrow \) Form 130 then follows.

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

Name Statement
\(\cal M2(\langle\omega_2\rangle)\) Feferman/Truss Model This is another extension of <a href="/models/Feferman-1">\(\cal M2\)</a>

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