Implications

This non-implication, Form 12 \( \not \Rightarrow \) Form 52, 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: 1262, whose string of implications is:
    43 \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 377 \(\Rightarrow\) 378 \(\Rightarrow\) 11 \(\Rightarrow\) 12
  • A proven non-implication whose code is 3. In this case, it's Code 3: 229, Form 43 \( \not \Rightarrow \) Form 223 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 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: 4197, whose string of implications is:
    52 \(\Rightarrow\) 221 \(\Rightarrow\) 222 \(\Rightarrow\) 223

The conclusion Form 12 \( \not \Rightarrow \) Form 52 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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