Implications

This non-implication, Form 93 \( \not \Rightarrow \) Form 16, 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: 10172, whose string of implications is:
    170 \(\Rightarrow\) 93
  • A proven non-implication whose code is 3. In this case, it's Code 3: 122, Form 170 \( \not \Rightarrow \) Form 6 whose summary information is:
    Hypothesis Statement
    Form 170 <p> \(\aleph_{1}\le 2^{\aleph_{0}}\). </p>

    Conclusion Statement
    Form 6 <p> \(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9876, whose string of implications is:
    16 \(\Rightarrow\) 6

The conclusion Form 93 \( \not \Rightarrow \) Form 16 then follows.

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

Name Statement
\(\cal M6\) Sageev's Model I Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)
\(\cal M36\) Figura's Model Starting with a countable, standard model, \(\cal M\), of \(ZFC + 2^{\aleph_0}=\aleph_{\omega +1}\), Figura uses forcing conditions that are functions from a subset of \(\omega\times\omega\) to \(\omega_\omega\) to construct a symmetric extension of \(\cal M\) in which there is an uncountable well ordered subset of the reals (<a href="/form-classes/howard-rubin-170">Form 170</a> is true), but \(\aleph_1= \aleph_{\omega}\) so \(\aleph_1\) is singular (<a href="/form-classes/howard-rubin-34">Form 34</a> is false)

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