Implications

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This non-implication, Form 119 \( \not \Rightarrow \) Form 76, 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: 7445, whose string of implications is: 115 \(\Rightarrow\) 118 \(\Rightarrow\) 119
• A proven non-implication whose code is 5. In this case, it's Code 3: 292, Form 115 \( \not \Rightarrow \) Form 76 whose summary information is:
  

  Hypothesis	Statement
  Form 115	<p> The product of weakly Loeb \(T_2\) spaces is weakly Loeb. <em>Weakly Loeb</em> means the set of non-empty closed subsets has a multiple choice function.) </p>

  
  

  Conclusion	Statement
  Form 76	<p> \(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\). </p>

  
  
• This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 119 \( \not \Rightarrow \) Form 76 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 \(A\)