Implications

This non-implication, Form 364 \( \not \Rightarrow \) Form 41, 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: 6902, whose string of implications is:
    91 \(\Rightarrow\) 363 \(\Rightarrow\) 364
  • A proven non-implication whose code is 5. In this case, it's Code 3: 216, Form 91 \( \not \Rightarrow \) Form 217 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

    Conclusion Statement
    Form 217 <p> Every infinite partially ordered set has either an infinite chain or an infinite antichain. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 906, whose string of implications is:
    41 \(\Rightarrow\) 9 \(\Rightarrow\) 217

The conclusion Form 364 \( \not \Rightarrow \) Form 41 then follows.

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

Name Statement
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).
\(\cal N34\) Pincus' Model X For each \(q\in\Bbb Q\), let \(C_q=\{a_q,b_q\}\), a pair of atoms and let \(A=\bigcup_{q\in\Bbb Q}C_q\)

Edit | Back