Implications

This non-implication, Form 83 \( \not \Rightarrow \) Form 49, 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: 10171, whose string of implications is:
    30 \(\Rightarrow\) 83
  • 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>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 83 \( \not \Rightarrow \) Form 49 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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