Implications

This non-implication, Form 293 \( \not \Rightarrow \) Form 345, 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: 10069, whose string of implications is:
    30 \(\Rightarrow\) 293
  • A proven non-implication whose code is 3. In this case, it's Code 3: 190, Form 30 \( \not \Rightarrow \) Form 49 whose summary information is:
    Hypothesis Statement
    Form 30 <p> <strong>Ordering Principle:</strong> Every set can be linearly ordered. </p>

    Conclusion Statement
    Form 49 <p> <strong>Order Extension Principle:</strong> Every partial ordering can be extended to a linear ordering.  <a href="/articles/Tarski-1924">Tarski [1924]</a>, p 78. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1392, whose string of implications is:
    345 \(\Rightarrow\) 14 \(\Rightarrow\) 49

The conclusion Form 293 \( \not \Rightarrow \) Form 345 then follows.

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

Name Statement
\(\cal M3\) Mathias' model Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\)
\(\cal M45\) Pincus' Model VII This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\)
\(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\)
\(\cal M14\) Morris' Model I This is an extension of Mathias' model, <a href="/models/Mathias-1">\(\cal M3\)</a>

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