Implications

This non-implication, Form 57 \( \not \Rightarrow \) Form 203, 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: 533, whose string of implications is:
    181 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 57
  • 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 57 \( \not \Rightarrow \) Form 203 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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