Implications

This non-implication, Form 165 \( \not \Rightarrow \) Form 343, 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: 7593, whose string of implications is:
    133 \(\Rightarrow\) 231 \(\Rightarrow\) 165
  • A proven non-implication whose code is 5. In this case, it's Code 3: 350, Form 133 \( \not \Rightarrow \) Form 146 whose summary information is:
    Hypothesis Statement
    Form 133  <p> Every set is either well orderable or has an infinite amorphous subset. </p>

    Conclusion Statement
    Form 146 <p> \(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is  an A1 space. (\((X,T)\) is A1 means if \(U \subseteq  T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4760, whose string of implications is:
    343 \(\Rightarrow\) 62 \(\Rightarrow\) 146

The conclusion Form 165 \( \not \Rightarrow \) Form 343 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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