Implications

This non-implication, Form 325 \( \not \Rightarrow \) Form 123, 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: 69, whose string of implications is:
    3 \(\Rightarrow\) 9 \(\Rightarrow\) 325
  • A proven non-implication whose code is 5. In this case, it's Code 3: 4, Form 3 \( \not \Rightarrow \) Form 344 whose summary information is:
    Hypothesis Statement
    Form 3  \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).

    Conclusion Statement
    Form 344 <p> If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9846, whose string of implications is:
    123 \(\Rightarrow\) 344

The conclusion Form 325 \( \not \Rightarrow \) Form 123 then follows.

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

Name Statement
\(\cal N9\) Halpern/Howard Model \(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\)

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