Implications

This non-implication, Form 57 \( \not \Rightarrow \) Form 113, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 192, Form 57 \( \not \Rightarrow \) Form 9 whose summary information is:
    Hypothesis Statement
    Form 57 <p> If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\). <br /> <a href="/articles/Mathias-1979">Mathias [1979]</a>, p 125. </p>

    Conclusion 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>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 279, whose string of implications is:
    113 \(\Rightarrow\) 8 \(\Rightarrow\) 9

The conclusion Form 57 \( \not \Rightarrow \) Form 113 then follows.

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

Name Statement
\(\cal M32\) Sageev's Model II Starting with a model \(\cal M\) of \(ZF + V =L\), Sageev constructs a sequence of models \(\cal M\subseteq N_0 \subseteq N_1\subseteq\cdots\subseteq N_{\kappa}\) where \(\kappa\) is an inaccessible cardinal, \(N_0\) is Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a>, and \(N_{\kappa}\) is \(\cal M32\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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