Implications

This non-implication, Form 6 \( \not \Rightarrow \) Form 86-alpha, 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: 5626, whose string of implications is:
    67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 337 \(\Rightarrow\) 92 \(\Rightarrow\) 94 \(\Rightarrow\) 6
  • A proven non-implication whose code is 5. In this case, it's Code 3: 156, Form 67 \( \not \Rightarrow \) Form 98 whose summary information is:
    Hypothesis Statement
    Form 67 <p> \(MC(\infty,\infty)\) \((MC)\), <strong>The Axiom of Multiple Choice:</strong> For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite). </p>

    Conclusion Statement
    Form 98 <p> The set of all finite subsets of a Dedekind finite set is Dedekind finite. <a href="/books/8">Jech [1973b]</a> p 161 prob 11.5. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 516, whose string of implications is:
    86-alpha \(\Rightarrow\) 8 \(\Rightarrow\) 9 \(\Rightarrow\) 98

The conclusion Form 6 \( \not \Rightarrow \) Form 86-alpha then follows.

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

Name Statement
\(\cal N2\) 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 \( A \).

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