Implications

This non-implication, Form 181 \( \not \Rightarrow \) Form 239, 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: 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: 7985, whose string of implications is:
    239 \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 91 \(\Rightarrow\) 79 \(\Rightarrow\) 203

The conclusion Form 181 \( \not \Rightarrow \) Form 239 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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