Implications

This non-implication, Form 309 \( \not \Rightarrow \) Form 14, 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)

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 5. In this case, it's Code 3: 603, Form 309 \( \not \Rightarrow \) Form 69 whose summary information is:
    Hypothesis Statement
    Form 309 <p> <strong>The Banach-Tarski Paradox:</strong> There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\). </p>

    Conclusion Statement
    Form 69 <p> Every field has an algebraic closure.  <a href="/books/8">Jech [1973b]</a>, p 13. <p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9602, whose string of implications is:
    14 \(\Rightarrow\) 69

The conclusion Form 309 \( \not \Rightarrow \) Form 14 then follows.

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

Name Statement
\(\cal N1\) The Basic Fraenkel Model The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\)

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