Implications

This non-implication, Form 351 \( \not \Rightarrow \) Form 181, 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: 2636, whose string of implications is:
    40 \(\Rightarrow\) 39 \(\Rightarrow\) 8 \(\Rightarrow\) 351
  • 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>

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

The conclusion Form 351 \( \not \Rightarrow \) Form 181 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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