Implications

This non-implication, Form 221 \( \not \Rightarrow \) Form 46-K, 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: 10226, whose string of implications is:
    63 \(\Rightarrow\) 221
  • A proven non-implication whose code is 3. In this case, it's Code 3: 887, Form 63 \( \not \Rightarrow \) Form 46-K whose summary information is:
    Hypothesis Statement
    Form 63 <p> \(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. <br /> <a href="/books/8">Jech [1973b]</a>, p 172 prob 8.5. </p>

    Conclusion Statement
    Form 46-K <p> If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function. </p>

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

The conclusion Form 221 \( \not \Rightarrow \) Form 46-K then follows.

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

Name Statement
\(\cal M47(n,M)\) Pincus' Model IX This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)

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