Axiom Of Choice

This non-implication, Form 70 \( \not \Rightarrow \) Form 8, 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: 10299, whose string of implications is: 63 \(\Rightarrow\) 70

A proven non-implication whose code is 5. In this case, it's Code 3: 136, Form 63 \( \not \Rightarrow \) Form 379 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 379	<p> \(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\). </p>

An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10238, whose string of implications is: 8 \(\Rightarrow\) 379

The conclusion Form 70 \( \not \Rightarrow \) Form 8 then follows.

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

Name	Statement
\(\cal N26\) Brunner/Pincus Model, a variation of \(\cal N2\)	The set ofatoms \(A=\bigcup_{n\in\omega} P_n\), where the \(P_n\)'s are pairwisedisjoint denumerable sets; \(\cal G\) is the set of all permutations\(\sigma\) on \(A\) such that \(\sigma(P_n)=P_n\), for all \(n\in\omega\); and \(S\)is the set of all finite subsets of \(A\)
\(\cal N29\) Dawson/Howard Model	Let \(A=\bigcup\{B_n; n\in\omega\}\) is a disjoint union, where each \(B_n\) is denumerable and ordered like the rationals by \(\le_n\)
\(\cal N49\) De la Cruz/Di Prisco Model	Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\)

Implications