Implications

This non-implication, Form 84 \( \not \Rightarrow \) Form 292, 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: 5915, whose string of implications is:
    77 \(\Rightarrow\) 185 \(\Rightarrow\) 84
  • 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: 6526, whose string of implications is:
    292 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 316

The conclusion Form 84 \( \not \Rightarrow \) Form 292 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 \(\Bbb R\) (\(|A| = 2^{\aleph_0}\))

Edit | Back