Implications

This non-implication, Form 268 \( \not \Rightarrow \) Form 50, whose code is 4, 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: 4009, whose string of implications is:
    14 \(\Rightarrow\) 49 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 268
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1, Form 14 \( \not \Rightarrow \) Form 50 whose summary information is:
    Hypothesis Statement
    Form 14 <p> <strong>BPI:</strong> Every Boolean algebra has a prime ideal. </p>

    Conclusion Statement
    Form 50 <p> <strong>Sikorski's  Extension Theorem:</strong> Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). <a href="/books/22">Sikorski [1964]</a>, p. 141. </p>

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

The conclusion Form 268 \( \not \Rightarrow \) Form 50 then follows.

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

Name Statement
\(\cal M1\) Cohen's original model Add a denumerable number of generic reals (subsets of \(\omega\)), \(a_1\), \(a_2\), \(\cdots\), along with the set \(b\) containing them

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