Implications

This non-implication, Form 44 \( \not \Rightarrow \) Form 261, 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: 153, Form 44 \( \not \Rightarrow \) Form 40 whose summary information is:
    Hypothesis Statement
    Form 44 <p> \(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\). </p>

    Conclusion Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8604, whose string of implications is:
    261 \(\Rightarrow\) 256 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40

The conclusion Form 44 \( \not \Rightarrow \) Form 261 then follows.

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

Name Statement
\(\cal N2(\aleph_{\alpha})\) Jech's Model This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)

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