Implications

This non-implication, Form 312 \( \not \Rightarrow \) Form 63, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1178, Form 312 \( \not \Rightarrow \) Form 223 whose summary information is:
    Hypothesis Statement
    Form 312 <p> A subgroup of an amenable group is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g \in G\), \(\mu(gA)=\mu(A)\).) </p>

    Conclusion Statement
    Form 223 <p> There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4862, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 206 \(\Rightarrow\) 223

The conclusion Form 312 \( \not \Rightarrow \) Form 63 then follows.

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

Name Statement
\(\cal M30\) Pincus/Solovay Model II In this construction, an \(\omega_1\) sequence of generic reals is added to a model of \(ZFC\) in such a way that the <strong>Principle of Dependent Choices</strong> (<a href="/form-classes/howard-rubin-43">Form 43</a>) is true, but no nonprincipal measure exists (<a href="/form-classes/howard-rubin-223">Form 223</a> is false)

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