Implications

This non-implication, Form 357 \( \not \Rightarrow \) Form 312, 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: 716, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 357
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1288, Form 43 \( \not \Rightarrow \) Form 312 whose summary information is:
    Hypothesis Statement
    Form 43 <p> \(DC(\omega)\) (DC), <strong>Principle of Dependent Choices:</strong> If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See <a href="/articles/Tarski-1948">Tarski [1948]</a>, p 96, <a href="/articles/Levy-1964">Levy [1964]</a>, p. 136. </p>

    Conclusion 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>

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

The conclusion Form 357 \( \not \Rightarrow \) Form 312 then follows.

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

Name Statement
\(\cal M5(\aleph)\) Solovay's Model An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)

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