Implications

This non-implication, Form 198 \( \not \Rightarrow \) Form 321, 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: 480, whose string of implications is:
    39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 198
  • A proven non-implication whose code is 3. In this case, it's Code 3: 54, Form 39 \( \not \Rightarrow \) Form 321 whose summary information is:
    Hypothesis Statement
    Form 39 <p> \(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p. 202. </p>

    Conclusion Statement
    Form 321 <p> There does not exist an ordinal \(\alpha\) such that \(\aleph_{\alpha}\) is weakly compact and \(\aleph_{\alpha+1}\) is measurable. </p>

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

The conclusion Form 198 \( \not \Rightarrow \) Form 321 then follows.

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

Name Statement
\(\cal M42\) Bull's Model Let \(\cal M\) be a countable transitive model of \(ZFC +\) "There are uncountable regular cardinals \(\aleph_\alpha <\aleph_\beta < \aleph_\gamma\) such that \(\aleph_\alpha\) is \(\aleph_\gamma\)-supercompact; \(\aleph_\beta\) is the first measurable cardinal greater than \(\aleph_\alpha\); and \(\aleph_\gamma =|2^{\aleph_\beta}|\)." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of \(\cal M\)

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