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December 31st, 2016, 07:23 PM   #41
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Originally Posted by v8archie View Post
... with addition and multiplication of the ordered pairs defined so as to preserve the bijection to the rational numbers ,,,
This is not informative.

Go ahead and define it.

In fact your example is not a bijection because $(1 , 4)$ and $(3 , 12)$ both map to the same rational $\frac{1}{4}$ yet they are clearly 2 different points in the Cartesian plane.

The reason Trichotomy is lost is because you are dealing with multi dimensional objects. For example , the following object $(1 , 4)$ as a point in the Cartesian plane is multi dimensional , in this case of dimension 2.

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