Implications

This non-implication, Form 419 \( \not \Rightarrow \) Form 253, 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: 3715, whose string of implications is:
    345 \(\Rightarrow\) 43 \(\Rightarrow\) 8 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 419
  • A proven non-implication whose code is 3. In this case, it's Code 3: 278, Form 345 \( \not \Rightarrow \) Form 253 whose summary information is:
    Hypothesis Statement
    Form 345 <p> <strong>Rasiowa-Sikorski Axiom:</strong>  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). </p>

    Conclusion Statement
    Form 253 <p> <strong>\L o\'s' Theorem:</strong> If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. </p>

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

The conclusion Form 419 \( \not \Rightarrow \) Form 253 then follows.

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

Name Statement
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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