Implications

This non-implication, Form 142 \( \not \Rightarrow \) Form 380, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4851, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 142
  • A proven non-implication whose code is 5. In this case, it's Code 3: 137, Form 63 \( \not \Rightarrow \) Form 380 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 380 <p> \(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function. </p>

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

The conclusion Form 142 \( \not \Rightarrow \) Form 380 then follows.

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

Name Statement
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)
\(\cal N49\) 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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