Implications

This non-implication, Form 313 \( \not \Rightarrow \) Form 346, 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: 6885, whose string of implications is:
    147 \(\Rightarrow\) 91 \(\Rightarrow\) 313
  • A proven non-implication whose code is 5. In this case, it's Code 3: 422, Form 147 \( \not \Rightarrow \) Form 126 whose summary information is:
    Hypothesis Statement
    Form 147 <p> \(A(D2)\):  Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets. </p>

    Conclusion Statement
    Form 126 <p> \(MC(\aleph_0,\infty)\), <strong>Countable axiom of multiple choice:</strong> For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10143, whose string of implications is:
    346 \(\Rightarrow\) 126

The conclusion Form 313 \( \not \Rightarrow \) Form 346 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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