Ordered Pair/Kuratowski Formalization/Warning

Warning

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

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


Sources

  • 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
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