Implications

This non-implication, Form 83 \( \not \Rightarrow \) 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: 9767, whose string of implications is:
    82 \(\Rightarrow\) 83
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1186, Form 82 \( \not \Rightarrow \) Form 253 whose summary information is:
    Hypothesis Statement
    Form 82 <p> \(E(I,III)\) (<a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) </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 83 \( \not \Rightarrow \) Form 253 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
\(\cal M40(\kappa)\) Pincus' Model IV The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\)
\(\cal N38\) Howard/Rubin Model I Let \((A,\le)\) be an ordered set of atomswhich is order isomorphic to \({\Bbb Q}^\omega\), the set of all functionsfrom \(\omega\) into \(\Bbb Q\) ordered by the lexicographic ordering
\(\cal N40\) Howard/Rubin Model II A variation of \(\cal N38\)

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