Implications

This non-implication, Form 31 \( \not \Rightarrow \) Form 132, whose code is 6, is constructed around a proven non-implication as follows:
Note: This non-implication is actually a code 4, as this non-implication satisfies the transferability criterion. Click Transfer details for all the details)

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1881, whose string of implications is:
    23 \(\Rightarrow\) 27 \(\Rightarrow\) 31
  • A proven non-implication whose code is 5. In this case, it's Code 3: 53, Form 23 \( \not \Rightarrow \) Form 132 whose summary information is:
    Hypothesis Statement
    Form 23 <p> \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). </p>

    Conclusion Statement
    Form 132 <p> \(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function. </p>

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

The conclusion Form 31 \( \not \Rightarrow \) Form 132 then follows.

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

Name Statement
\(\cal N24\) Hickman's Model I This model is a variation of \(\cal N2\)

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