Implications

This non-implication, Form 18 \( \not \Rightarrow \) Form 69, whose code is 6, 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: 2344, whose string of implications is:
    16 \(\Rightarrow\) 352 \(\Rightarrow\) 31 \(\Rightarrow\) 32 \(\Rightarrow\) 10 \(\Rightarrow\) 80 \(\Rightarrow\) 18
  • A proven non-implication whose code is 5. In this case, it's Code 3: 11, Form 16 \( \not \Rightarrow \) Form 69 whose summary information is:
    Hypothesis Statement
    Form 16 <p> \(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function. </p>

    Conclusion Statement
    Form 69 <p> Every field has an algebraic closure.  <a href="/books/8">Jech [1973b]</a>, p 13. <p>

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

The conclusion Form 18 \( \not \Rightarrow \) Form 69 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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