Implications

This non-implication, Form 273 \( \not \Rightarrow \) Form 333, 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: 6882, whose string of implications is:
    147 \(\Rightarrow\) 91 \(\Rightarrow\) 273
  • A proven non-implication whose code is 5. In this case, it's Code 3: 418, Form 147 \( \not \Rightarrow \) Form 67 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 67 <p> \(MC(\infty,\infty)\) \((MC)\), <strong>The Axiom of Multiple Choice:</strong> For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9913, whose string of implications is:
    333 \(\Rightarrow\) 67

The conclusion Form 273 \( \not \Rightarrow \) Form 333 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 \(A\)

Edit | Back