Implications

This non-implication, Form 150 \( \not \Rightarrow \) Form 213, 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: 9940, whose string of implications is:
    85 \(\Rightarrow\) 150
  • A proven non-implication whose code is 3. In this case, it's Code 3: 224, Form 85 \( \not \Rightarrow \) Form 213 whose summary information is:
    Hypothesis Statement
    Form 85 <p> \(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  <a href="/books/8">Jech [1973b]</a> p 115 prob 7.13. </p>

    Conclusion Statement
    Form 213 <p> \(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function. </p>

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

The conclusion Form 150 \( \not \Rightarrow \) Form 213 then follows.

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

Name Statement
\(\cal M34(\aleph_1)\) Pincus' Model III Pincus proves that Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a> can be extended by adding \(\aleph_1\) generic sets along with the set \(b\) containing them and well orderings of all countable subsets of \(b\)

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