Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 411, 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: 215, Form 0 \( \not \Rightarrow \) Form 10 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 10 <p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9570, whose string of implications is:
    411 \(\Rightarrow\) 412 \(\Rightarrow\) 10

The conclusion Form 0 \( \not \Rightarrow \) Form 411 then follows.

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

Name Statement
\(\cal M4\) Pincus' Model I This model has many of the properties of the model given in <a href="/books/39">Cohen [1966]</a> (p 143)
\(\cal M7\) Cohen's Second Model There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model
\(\cal N2\) The Second Fraenkel Model The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \).
\(\cal N2(n)\) 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, \(T_i = \{a_i, b_i,c_i\}\)
\(\cal N6\) 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
\(\cal N22(p)\) 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 \(p\)
\(\cal N35\) Truss' Model IV The set of atoms, \(A\), is denumerable andeach element of \(A\) is associated with a finite sequence of zeros andones
\(\cal N43\) Brunner's Model II The set of atoms \(A=\bigcup\{P_n: n\in\omega\}\), where \(|P_n|=n+1\) for each \(n\in\omega\) and the \(P_n\)'s arepairwise disjoint
\(\cal N50(E)\) Brunner's Model III \(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)

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