Implications

This non-implication, Form 356 \( \not \Rightarrow \) Form 256, 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: 10272, whose string of implications is:
    85 \(\Rightarrow\) 356
  • A proven non-implication whose code is 3. In this case, it's Code 3: 20, Form 85 \( \not \Rightarrow \) Form 78 whose summary information is:
    Hypothesis Statement
    Form 85 <p> \(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  <a href="/books/8">Jech [1973b]</a> p 115 prob 7.13. </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: 8238, whose string of implications is:
    256 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 43 \(\Rightarrow\) 78

The conclusion Form 356 \( \not \Rightarrow \) Form 256 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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