Implications

This non-implication, Form 70 \( \not \Rightarrow \) Form 262, 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: 10299, whose string of implications is:
    63 \(\Rightarrow\) 70
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1076, Form 63 \( \not \Rightarrow \) Form 151 whose summary information is:
    Hypothesis Statement
    Form 63 <p> \(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. <br /> <a href="/books/8">Jech [1973b]</a>, p 172 prob 8.5. </p>

    Conclusion Statement
    Form 151 <p> \(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see <a href="/notes/note-27">note 27</a> for \(UT(WO,\kappa,WO)\).) </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 8510, whose string of implications is:
    262 \(\Rightarrow\) 255 \(\Rightarrow\) 260 \(\Rightarrow\) 40 \(\Rightarrow\) 231 \(\Rightarrow\) 151

The conclusion Form 70 \( \not \Rightarrow \) Form 262 then follows.

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

Name Statement
\(\cal M47(n,M)\) Pincus' Model IX This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((E)\)
\(\cal N2(\aleph_{\alpha})\) Jech's Model This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)
\(\cal N15\) Brunner/Howard Model I \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\)
\(\cal N49\) De la Cruz/Di Prisco Model Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

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