Implications

This non-implication, Form 134 \( \not \Rightarrow \) Form 1, whose code is 6, 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 5. In this case, it's Code 3: 364, Form 134 \( \not \Rightarrow \) Form 69 whose summary information is:
    Hypothesis Statement
    Form 134 <p> If \(X\) is an infinite \(T_1\) space and \(X^{Y}\) is \(T_5\), then \(Y\) is countable. (\(T_5\) is 'hereditarily \(T_4\)'.) </p>

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

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10383, whose string of implications is:
    1 \(\Rightarrow\) 69

The conclusion Form 134 \( \not \Rightarrow \) Form 1 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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