Ordered Pair/Kuratowski Formalization/Warning

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Warning

The weakness of the Kuratowski formalization of the ordered pair shows up when $a = b$:

\(\displaystyle \tuple {a, a}\) \(=\) \(\displaystyle \set {\set a, \set {a, a} }\) Definition of Kuratowski Formalization of Ordered Pair
\(\displaystyle \) \(=\) \(\displaystyle \set {\set a, \set a}\) Definition of Uniqueness of Set Elements
\(\displaystyle \) \(=\) \(\displaystyle \set {\set a}\) Definition of Uniqueness of Set Elements

Thus the ordered pair degenerates into the set $\set {\set a}$.

Most works on this subject gloss over this point, and indeed, completely fail to mention it.


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