Implications

This non-implication, Form 127 \( \not \Rightarrow \) Form 379, 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: 4875, whose string of implications is:
    11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n \(\Rightarrow\) 64 \(\Rightarrow\) 127
  • A proven non-implication whose code is 3. In this case, it's Code 3: 743, Form 11 \( \not \Rightarrow \) Form 379 whose summary information is:
    Hypothesis Statement
    Form 11 <p> <strong>A Form of Restricted Choice for Families of Finite Sets:</strong> For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. <a href="/excerpts/De-la-cruz-di-prisco-1998b">De la Cruz/Di Prisco [1998b]</a> </p>

    Conclusion Statement
    Form 379 <p> \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). </p>

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

The conclusion Form 127 \( \not \Rightarrow \) Form 379 then follows.

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

Name Statement
\(\cal N6\) Levy's Model I \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th prime
\(\cal N29\) Dawson/Howard Model Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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