Implications

This non-implication, Form 222 \( \not \Rightarrow \) Form 62, 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: 4852, whose string of implications is:
    63 \(\Rightarrow\) 70 \(\Rightarrow\) 222
  • A proven non-implication whose code is 5. In this case, it's Code 3: 125, Form 63 \( \not \Rightarrow \) Form 146 whose summary information is:
    Hypothesis Statement
    Form 63 <p> \(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. <br /> <a href="/books/8">Jech [1973b]</a>, p 172 prob 8.5. </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: 9628, whose string of implications is:
    62 \(\Rightarrow\) 146

The conclusion Form 222 \( \not \Rightarrow \) Form 62 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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