Implications

This non-implication, Form 208 \( \not \Rightarrow \) Form 407, 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: 3310, whose string of implications is:
    202 \(\Rightarrow\) 40 \(\Rightarrow\) 208
  • A proven non-implication whose code is 5. In this case, it's Code 3: 537, Form 202 \( \not \Rightarrow \) Form 344 whose summary information is:
    Hypothesis Statement
    Form 202 <p> \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function. </p>

    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: 1495, whose string of implications is:
    407 \(\Rightarrow\) 14 \(\Rightarrow\) 123 \(\Rightarrow\) 344

The conclusion Form 208 \( \not \Rightarrow \) Form 407 then follows.

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

Name Statement
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false

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