Implications

This non-implication, Form 271-n \( \not \Rightarrow \) Form 99, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4228, whose string of implications is:
    60 \(\Rightarrow\) 85 \(\Rightarrow\) 270 \(\Rightarrow\) 271-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 181, Form 60 \( \not \Rightarrow \) Form 99 whose summary information is:
    Hypothesis Statement
    Form 60 <p> \(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.<br /> <a href="/books/2">Moore, G. [1982]</a>, p 125. </p>

    Conclusion Statement
    Form 99 <p> <strong>Rado's Selection Lemma:</strong> Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. </p>

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

The conclusion Form 271-n \( \not \Rightarrow \) Form 99 then follows.

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

Name Statement
\(\cal M3\) Mathias' model Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)
\(\cal N5\) 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 \(A\)
\(\cal N7\) L\"auchli's Model I \(A\) is countably infinite

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