Implications

This non-implication, Form 336-n \( \not \Rightarrow \) Form 171, 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: 1273, whose string of implications is:
    11 \(\Rightarrow\) 12 \(\Rightarrow\) 336-n
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1122, Form 11 \( \not \Rightarrow \) Form 171 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 171 <p> If \((P,\le)\) is a partial order such that \(P\) is the denumerable union of finite sets and all antichains in \(P\) are finite then for each denumerable family \({\cal D}\) of dense sets there is a \({\cal D}\) generic filter. </p>

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

The conclusion Form 336-n \( \not \Rightarrow \) Form 171 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

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