Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 10, 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: 10877, whose string of implications is:
    111 \(\Rightarrow\) 0
  • A proven non-implication whose code is 5. In this case, it's Code 3: 254, Form 111 \( \not \Rightarrow \) Form 288-n whose summary information is:
    Hypothesis Statement
    Form 111 <p> \(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. </p>

    Conclusion Statement
    Form 288-n <p> If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10286, whose string of implications is:
    10 \(\Rightarrow\) 288-n

The conclusion Form 0 \( \not \Rightarrow \) Form 10 then follows.

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

Name Statement
\(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)

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