Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 106, 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: 929, Form 165 \( \not \Rightarrow \) Form 78 whose summary information is:
    Hypothesis Statement
    Form 165 <p> \(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function. </p>

    Conclusion Statement
    Form 78 <p> <strong>Urysohn's Lemma:</strong>  If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). <a href="/articles/Urysohn-1925">Urysohn [1925]</a>, pp 290-292. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9610, whose string of implications is:
    106 \(\Rightarrow\) 78

The conclusion Form 165 \( \not \Rightarrow \) Form 106 then follows.

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

Name Statement
\(\cal N3\) Mostowski's Linearly Ordered Model \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\)

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