Implications

This non-implication, Form 211 \( \not \Rightarrow \) Form 214, 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: 10221, whose string of implications is:
    43 \(\Rightarrow\) 211
  • A proven non-implication whose code is 3. In this case, it's Code 3: 64, Form 43 \( \not \Rightarrow \) Form 173 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 173 <p> \(MPL\): Metric spaces are para-Lindel&ouml;f. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5912, whose string of implications is:
    214 \(\Rightarrow\) 76 \(\Rightarrow\) 173

The conclusion Form 211 \( \not \Rightarrow \) Form 214 then follows.

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

Name Statement
\(\cal N57\) The set of atoms \(A=\cup\{A_{n}:n\in\aleph_{1}\}\), where\(A_{n}=\{a_{nx}:x\in B(0,1)\}\) and \(B(0,1)\) is the set of points on theunit circle centered at 0 The group of permutations \(\cal{G}\) is thegroup of all permutations on \(A\) which rotate the \(A_{n}\)'s by an angle\(\theta_{n}\in\Bbb{R}\) and supports are countable

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