Implications

This non-implication, Form 285 \( \not \Rightarrow \) Form 303, 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: 2078, whose string of implications is:
    49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 285
  • A proven non-implication whose code is 5. In this case, it's Code 3: 118, Form 49 \( \not \Rightarrow \) Form 14 whose summary information is:
    Hypothesis Statement
    Form 49 <p> <strong>Order Extension Principle:</strong> Every partial ordering can be extended to a linear ordering.  <a href="/articles/Tarski-1924">Tarski [1924]</a>, p 78. </p>

    Conclusion Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 4051, whose string of implications is:
    303 \(\Rightarrow\) 50 \(\Rightarrow\) 14

The conclusion Form 285 \( \not \Rightarrow \) Form 303 then follows.

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

Name Statement
\(\cal N52\) Felgner/Truss Model Let \((\cal B,\prec)\) be a countableuniversal homogeneous linearly ordered Boolean algebra, (i.e., \(<\) is alinear ordering extending the Boolean partial ordering on \(B\))

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