Implications

This non-implication, Form 155 \( \not \Rightarrow \) Form 87-alpha, 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: 3946, whose string of implications is:
    43 \(\Rightarrow\) 78 \(\Rightarrow\) 155
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1141, Form 43 \( \not \Rightarrow \) Form 183-alpha whose summary information is:
    Hypothesis Statement
    Form 43 <p> \(DC(\omega)\) (DC), <strong>Principle of Dependent Choices:</strong> If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See <a href="/articles/Tarski-1948">Tarski [1948]</a>, p 96, <a href="/articles/Levy-1964">Levy [1964]</a>, p. 136. </p>

    Conclusion Statement
    Form 183-alpha <p> There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that <ol type="1"> <li>\(|X|\) is incomparable with \(\aleph_{\alpha}\)</li> <li>\(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and</li> <li>\(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).</li> </ol> </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5904, whose string of implications is:
    87-alpha \(\Rightarrow\) 71-alpha \(\Rightarrow\) 183-alpha

The conclusion Form 155 \( \not \Rightarrow \) Form 87-alpha then follows.

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

Name Statement
\(\cal N27\) Hickman's Model II Let \(A\) be a set with cardinality\(\aleph_1\) such that \(A=\{(a_{\alpha},b_{\beta}) : \alpha < \omega, \beta< \omega_1\}\)

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