Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 136-k, 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: 9695, whose string of implications is:
    91 \(\Rightarrow\) 309
  • A proven non-implication whose code is 5. In this case, it's Code 3: 203, Form 91 \( \not \Rightarrow \) Form 136-k 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 136-k <p> <strong>Surjective Cardinal Cancellation</strong> (depends on \(k\in\omega-\{0\}\)): For all cardinals \(x\) and \(y\), \(kx\le^* ky\) implies \(x\le^* y\). </p>

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

The conclusion Form 309 \( \not \Rightarrow \) Form 136-k then follows.

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

Name Statement
\(\cal N20\) Truss' Model II <p> Let \(X=\{a(i,k,l): i\in 2, k\in \Bbb Z, l\in\omega\}\), \(Y=\{a(i,j,k,l): i,j\in 2, k\in\Bbb Z, i\in\omega\}\) and \(A\) is the disjoint union of \(X\) and \(Y\) </p>

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