Implications

This non-implication, Form 268 \( \not \Rightarrow \) Form 21, 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: 4340, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 62 \(\Rightarrow\) 61 \(\Rightarrow\) 88 \(\Rightarrow\) 268
  • A proven non-implication whose code is 3. In this case, it's Code 3: 853, Form 295 \( \not \Rightarrow \) Form 5 whose summary information is:
    Hypothesis Statement
    Form 295 <p> <strong>DO:</strong>  Every infinite set has a dense linear ordering. </p>

    Conclusion Statement
    Form 5 <p> \(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1856, whose string of implications is:
    21 \(\Rightarrow\) 23 \(\Rightarrow\) 27 \(\Rightarrow\) 31 \(\Rightarrow\) 6 \(\Rightarrow\) 5

The conclusion Form 268 \( \not \Rightarrow \) Form 21 then follows.

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

Name Statement
\(\cal M6\) Sageev's Model I Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\)

Edit | Back