Implications

This non-implication, Form 222 \( \not \Rightarrow \) Form 259, 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: 4166, whose string of implications is:
    259 \(\Rightarrow\) 51 \(\Rightarrow\) 294

The conclusion Form 222 \( \not \Rightarrow \) Form 259 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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