Implications

This non-implication, Form 19 \( \not \Rightarrow \) Form 27, 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: 2347, whose string of implications is:
    31 \(\Rightarrow\) 34 \(\Rightarrow\) 19
  • A proven non-implication whose code is 3. In this case, it's Code 3: 874, Form 31 \( \not \Rightarrow \) Form 27 whose summary information is:
    Hypothesis Statement
    Form 31 <p>\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): <strong>The countable union theorem:</strong>  The union of a denumerable set of denumerable sets is denumerable. </p>

    Conclusion Statement
    Form 27 <p> \((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). <a href="/books/2">Moore, G. [1982]</a>, p 36. </p>

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

The conclusion Form 19 \( \not \Rightarrow \) Form 27 then follows.

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

Name Statement
\(\cal N17\) Brunner/Howard Model II \(A=\{a_{\alpha,i}:\alpha\in\omega_1\,\wedge i\in\omega\}\)

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