Implications

This non-implication, Form 339 \( \not \Rightarrow \) Form 264, 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: 3334, whose string of implications is:
    40 \(\Rightarrow\) 43 \(\Rightarrow\) 339
  • A proven non-implication whose code is 3. In this case, it's Code 3: 79, Form 40 \( \not \Rightarrow \) Form 181 whose summary information is:
    Hypothesis Statement
    Form 40 <p> \(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. <a href="/books/2">Moore, G. [1982]</a>, p 325. </p>

    Conclusion Statement
    Form 181 <p> \(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7861, whose string of implications is:
    264 \(\Rightarrow\) 202 \(\Rightarrow\) 181

The conclusion Form 339 \( \not \Rightarrow \) Form 264 then follows.

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

Name Statement
\(\cal M25\) Freyd's Model Using topos-theoretic methods due to Fourman, Freyd constructs a Boolean-valued model of \(ZF\) in which every well ordered family of sets has a choice function (<a href="/form-classes/howard-rubin-40">Form 40</a> is true), but \(C(|\Bbb R|,\infty)\) (<a href="/form-classes/howard-rubin-181">Form 181</a>) is false

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