Implications

This non-implication, Form 185 \( \not \Rightarrow \) Form 51, 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: 9707, whose string of implications is:
    77 \(\Rightarrow\) 185
  • A proven non-implication whose code is 5. In this case, it's Code 3: 168, Form 77 \( \not \Rightarrow \) Form 316 whose summary information is:
    Hypothesis 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>

    Conclusion Statement
    Form 316 <p> If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\)  has the <em>fixed point property</em> if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is <em>complete</em> if every subset of \(A\) has a least upper bound.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9704, whose string of implications is:
    51 \(\Rightarrow\) 316

The conclusion Form 185 \( \not \Rightarrow \) Form 51 then follows.

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

Name Statement
\(\cal N47\) H&ouml;ft/Howard Model II This model is similar to \(\cal N33\).The atoms \(A\) are ordered by \(\le\) so that they have order type that ofthe real numbers \(\Bbb R\) (\(|A| = 2^{\aleph_0}\))

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