Implications

This non-implication, Form 12 \( \not \Rightarrow \) Form 303, 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: 1253, whose string of implications is:
    3 \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 377 \(\Rightarrow\) 378 \(\Rightarrow\) 11 \(\Rightarrow\) 12
  • 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: 4091, whose string of implications is:
    303 \(\Rightarrow\) 50 \(\Rightarrow\) 14 \(\Rightarrow\) 123 \(\Rightarrow\) 344

The conclusion Form 12 \( \not \Rightarrow \) Form 303 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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