Implications

This non-implication, Form 31 \( \not \Rightarrow \) 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: 1708, whose string of implications is:
    16 \(\Rightarrow\) 352 \(\Rightarrow\) 31
  • A proven non-implication whose code is 5. In this case, it's Code 3: 12, Form 16 \( \not \Rightarrow \) Form 84 whose summary information is:
    Hypothesis Statement
    Form 16 <p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function. </p>

    Conclusion Statement
    Form 84 <p> \(E(II,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 891, whose string of implications is:
    41 \(\Rightarrow\) 9 \(\Rightarrow\) 84

The conclusion Form 31 \( \not \Rightarrow \) Form 41 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)
\(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\)

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