TITLE: Definition of the ordered triple (a, b, c) according to Kuratowski's Set Theory.
QUESTION [5 upvotes]: Can someone give Kuratowski's definition of the ordered triple $(a,b,c)$ assuming $A \times B \times C$ is rewritten as $(A \times B) \times C$, please? I noticed there is already an answered question for the ordered $n$-tuple, but (as I'm very new to Maths) I didn't understand it, and I only need the definition for the ordered triple.

REPLY [4 votes]: I know I come a litle late but wanted to point something out. It has been said that you can define a triple as
$$(a,b,c) = \{\{a\},\{a,b\},\{a,b,c\}\}$$
but this definition has a big problem as you can see that for any a,b,c it holds that $(a,b,a)=(a,b,b)$, as they're both equal to $\{\{a\},\{a,b\}\}$.
This is because you haven't paid attention to one extremely important property that you want any ordered pair (or n-tuple really) to have, which is
$$(x,y)=(x',y') \Leftrightarrow x=x',y=y'$$
I believe that's why we define n-tuples recursively and not the way the former comment suggested.