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Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which captures the intuition, that an ordered pair has a first and a second element.

Membership graphs for possible definitions of ordered pairs (≙ top node, arrow heads omitted)

1: (x,y) := {   x   , {   x   ,   y } }
2: (x,y) := { { x } , {   x   ,   y } } (Kuratowski's definition)
3: (x,y) := { { x } , { { x } ,   y } }
4: (x,y) := { { x , 0 }  ,  { 1 , y } } (Hausdorff's definition)


So my question is:

Are there good reasons to choose Kuratowski's definition (or did Kuratowski himself give any) instead of one of the more "elegant" - sparing, symmetric, or intuitive - alternatives?

1

Kuratowski's definition of ordered pairs

Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which captures the intuition, that an ordered pair has a first and a second element.

Membership graphs for possible definitions of ordered pairs (≙ top node, arrow heads omitted)

1: (x,y) := {   x   , {   x   ,   y } }
2: (x,y) := { { x } , {   x   ,   y } } (Kuratowski's definition)
3: (x,y) := { { x } , { { x } ,   y } }
4: (x,y) := { { x , 0 }  ,  { 1 , y } }


So my question is:

Are there good reasons to choose Kuratowski's definition (or did Kuratowski himself give any) instead of one of the more "elegant" - sparing, symmetric, or intuitive - alternatives?