Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 422-n, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 57, Form 0 \( \not \Rightarrow \) Form 47-n whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 47-n <p> If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection 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: 10251, whose string of implications is:
    422-n \(\Rightarrow\) 47-n

The conclusion Form 0 \( \not \Rightarrow \) Form 422-n then follows.

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

Name Statement
\(\cal M7(n)\) Generalization of \(\cal M7\) Model <a href="/models/Cohen-2">\(\cal M7\)</a> can be generalized to \(n\) denumerable sets for \(1 \le n \in\omega\), then the Axiom of Choice for a denumerable number of \(n\) element sets, \(C(\aleph_0,n)\), is false for \(1 \le n \le \omega\)
\(\cal M47(n,M)\) Pincus' Model IX This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)
\(\cal N2(n)\) A generalization of \(\cal N2\) This is a generalization of\(\cal N2\) in which there is a denumerable set of \(n\) element sets for\(n\in\omega - \{0,1\}\)
\(\cal N2^*(3)\) Howard's variation of \(\cal N2(3)\) \(A=\bigcup B\), where\(B\) is a set of pairwise disjoint 3 element sets, \(T_i = \{a_i, b_i,c_i\}\)
\(\cal N2(n,M)\) Mostowski's variation of \(\cal N2(n)\) \(A\), \(B\), and \(S\)are the same as in \(\cal N2(n)\)
\(\cal N22(p)\) Makowski/Wi\'sniewski/Mostowski Model (Where \(p\) is aprime) Let \(A=\bigcup\{A_i: i\in\omega\}\) where The \(A_i\)'s are pairwisedisjoint and each has cardinality \(p\)
\(\cal N49\) 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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