Implications

This non-implication, Form 223 \( \not \Rightarrow \) Form 345, 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: 4862, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 206 \(\Rightarrow\) 223
  • A proven non-implication whose code is 3. In this case, it's Code 3: 892, Form 63 \( \not \Rightarrow \) Form 47-n 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 47-n <p> If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2236, whose string of implications is:
    345 \(\Rightarrow\) 14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 121 \(\Rightarrow\) 122 \(\Rightarrow\) 47-n

The conclusion Form 223 \( \not \Rightarrow \) Form 345 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 \((E)\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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