Implications

This non-implication, Form 307 \( \not \Rightarrow \) Form 96, 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: 6895, whose string of implications is:
    91 \(\Rightarrow\) 305 \(\Rightarrow\) 307
  • A proven non-implication whose code is 5. In this case, it's Code 3: 195, Form 91 \( \not \Rightarrow \) Form 96 whose summary information is:
    Hypothesis Statement
    Form 91 <p> \(PW\):  The power set of a well ordered set can be well ordered. </p>

    Conclusion Statement
    Form 96 <p> <strong>L&ouml;wig's Theorem:</strong>If \(B_{1}\) and \(B_{2}\) are both bases for the vector space \(V\) then \(|B_{1}| = |B_{2}|\). </p>

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

The conclusion Form 307 \( \not \Rightarrow \) Form 96 then follows.

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

Name Statement
\(\cal N13\) L\"auchli/Jech Model \(A = B_1\cup B_2\), where \(B_1=\bigcup\{A_{j1} : j\in\omega\}\), and \(B_2 = \bigcup\{A_{j2} :j\in\omega\}\), and each \(A_{ji}\) is a 6-element set

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