Implications

This non-implication, Form 43 \( \not \Rightarrow \) Form 239, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 275, Form 43 \( \not \Rightarrow \) Form 144 whose summary information is:
    Hypothesis Statement
    Form 43 <p> \(DC(\omega)\) (DC), <strong>Principle of Dependent Choices:</strong> If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See <a href="/articles/Tarski-1948">Tarski [1948]</a>, p 96, <a href="/articles/Levy-1964">Levy [1964]</a>, p. 136. </p>

    Conclusion Statement
    Form 144 <p> Every set is almost well orderable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7896, whose string of implications is:
    239 \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 144

The conclusion Form 43 \( \not \Rightarrow \) Form 239 then follows.

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

Name Statement
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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