Implications

This non-implication, Form 88 \( \not \Rightarrow \) Form 355, 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: 10254, whose string of implications is:
    333 \(\Rightarrow\) 88
  • A proven non-implication whose code is 5. In this case, it's Code 3: 666, Form 333 \( \not \Rightarrow \) Form 358 whose summary information is:
    Hypothesis Statement
    Form 333 <p> \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd. </p>

    Conclusion Statement
    Form 358 <p> \(KW(\aleph_0,<\aleph_0)\), <strong>The Kinna-Wagner Selection Principle</strong> for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10012, whose string of implications is:
    355 \(\Rightarrow\) 358

The conclusion Form 88 \( \not \Rightarrow \) Form 355 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, \(T_i = \{a_i, b_i,c_i\}\)
\(\cal N6\) 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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