Implications

This non-implication, Form 113 \( \not \Rightarrow \) Form 43, whose code is 6, 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 5. In this case, it's Code 3: 270, Form 113 \( \not \Rightarrow \) Form 43 whose summary information is:
    Hypothesis Statement
    Form 113 <p> <strong>Tychonoff's Compactness Theorem for Countably Many Spaces:</strong> The product of a countable set of compact spaces is compact. </p>

    Conclusion 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>

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

The conclusion Form 113 \( \not \Rightarrow \) Form 43 then follows.

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

Name Statement
\(\cal N38\) Howard/Rubin Model I Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering

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