Implications

This non-implication, Form 328 \( \not \Rightarrow \) Form 391, 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: 3306, whose string of implications is:
    202 \(\Rightarrow\) 40 \(\Rightarrow\) 328
  • A proven non-implication whose code is 5. In this case, it's Code 3: 536, Form 202 \( \not \Rightarrow \) Form 323 whose summary information is:
    Hypothesis Statement
    Form 202 <p> \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function. </p>

    Conclusion Statement
    Form 323 <p> \(KW(\infty,WO)\), <strong>The Kinna-Wagner Selection Principle for a family of well orderable sets:</strong> For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See <a href="/form-classes/howard-rubin-15">Form 15</a>.) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9458, whose string of implications is:
    391 \(\Rightarrow\) 399 \(\Rightarrow\) 323

The conclusion Form 328 \( \not \Rightarrow \) Form 391 then follows.

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

Name Statement
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false

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