Implications

This non-implication, Form 390 \( \not \Rightarrow \) Form 117, 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: 745, whose string of implications is:
    39 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 64 \(\Rightarrow\) 390
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7448, whose string of implications is:
    117 \(\Rightarrow\) 319 \(\Rightarrow\) 320 \(\Rightarrow\) 321

The conclusion Form 390 \( \not \Rightarrow \) Form 117 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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