Implications

This non-implication, Form 222 \( \not \Rightarrow \) Form 89, 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: 4852, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 222
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1250, Form 63 \( \not \Rightarrow \) Form 294 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 294 <p> Every linearly ordered \(W\)-set is well orderable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6399, whose string of implications is:
    89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 294

The conclusion Form 222 \( \not \Rightarrow \) Form 89 then follows.

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

Name Statement
\(\cal N41\) Another variation of \(\cal N3\) \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\)

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