Implications

This non-implication, Form 124 \( \not \Rightarrow \) Form 87-alpha, 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: 1019, whose string of implications is:
    9 \(\Rightarrow\) 17 \(\Rightarrow\) 124
  • A proven non-implication whose code is 3. In this case, it's Code 3: 927, Form 9 \( \not \Rightarrow \) Form 77 whose summary information is:
    Hypothesis Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

    Conclusion Statement
    Form 77 <p> A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. <a href="/books/8">Jech [1973b]</a>, p 23. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 3938, whose string of implications is:
    87-alpha \(\Rightarrow\) 43 \(\Rightarrow\) 77

The conclusion Form 124 \( \not \Rightarrow \) Form 87-alpha then follows.

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

Name Statement
\(\cal N10\) H&ouml;ft/Howard/Mostowski Model (The model is a variation of\(\cal N3\).) \(A\) as ordered by \(\precsim\) has the same order type as therationals; \(\cal G\) is the group of all order automorphisms of \(A\); \(S\) isthe set of all subsets \(E\) of \(A\) that satisfy the following threeconditions:\item{1.} \(E\) is well ordered by \(\precsim\).\item{2.} \(E\) is bounded in \(A\).\item{3.} If \(b:\alpha\to E\) is an order preserving bijection from\(\alpha\) onto \(E\) and if \(\lambda < \alpha\) is a limit ordinal then\(\{b(\beta) : \beta < \gamma\}\) has no least upper bound in\((A,\precsim)\).\par\noindentIn <a href="/articles/H\"oft/Howard-1994">H\"oft/Howard [1994]</a> it is shown that, in \(\cal N10\), everyDedekind finite set is finite (9 is true), but \((A,\precsim)\) is alinearly ordered set with no infinite descending sequences that cannot bewell ordered (77 is false)

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