Implications

This non-implication, Form 18 \( \not \Rightarrow \) Form 110, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 6213, whose string of implications is:
    111 \(\Rightarrow\) 80 \(\Rightarrow\) 18
  • A proven non-implication whose code is 3. In this case, it's Code 3: 170, Form 111 \( \not \Rightarrow \) Form 110 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 110 <p> Every vector space over \(\Bbb Q\) has a basis. </p>

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

The conclusion Form 18 \( \not \Rightarrow \) Form 110 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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