Implications

This non-implication, Form 341 \( \not \Rightarrow \) Form 264, 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: 287, whose string of implications is:
    113 \(\Rightarrow\) 8 \(\Rightarrow\) 341
  • 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7789, whose string of implications is:
    264 \(\Rightarrow\) 202 \(\Rightarrow\) 40 \(\Rightarrow\) 43

The conclusion Form 341 \( \not \Rightarrow \) Form 264 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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