One issue about Kuratowski definition of an ordered pair.

[cheesecakes]

Hello.

I have understood the Kuratowski definition of the ordered pair and appreciate it's usefulness but have a nagging difficulty about it.

Consider an ordered pair which is (a,a). according to Kuratowski definition it is defined as {{a},{a,a}} . Now consider an ordered triplet (a,a,a) it would be defined as {{a},{a,a},{a,a,a}}.

My point is isn't {{a},{a,a}} same as {a}
and isn't {{a},{a,a},{a,a,a}} also same as {a} .

So how to distinguish between (a,a) and (a,a,a) using Kuratowski definition?I am painfully aware that I am missing out on some basic set theory fundamental over here.
Is it implicit that when we use sets to define mathematical objects we restrain ourselves to that particular object only.As in this case when we define ordered pairs as sets we have it as an implicit assumption that this set is an ordered pair??

 

[Hurkyl]

Now consider an ordered triplet (a,a,a) it would be defined as {{a},{a,a},{a,a,a}}.

You are considering defining ordered triples as

(a,b,c)={{a}, {a,b}, {a,b,c}}​

then? I've never seen such a definition proposed -- in fact, it is definitely inadequate because, by that definition,

(a,b,b) = (a,a,b)​

That aside, typically you are never in a situation where you are wondering if some mathematical object is an ordered pair or an ordered triple. In the odd case that you needed an collection that contained objects of both types, then if necessary one would put into the collection extra information to allow you to tell the difference (assuming you really did need to be able to do so).

 

[disregardthat]

The definition of a triple is (a,b,c) := (a,(b,c)). Since (a,a) = {{a},{a,a}} = {{a},{a}} = {{a}}, we will have that (a,a,a) = (a,(a,a)) = (a,{{a}}) =/= (a,a) by uniqueness of components.

Normally when you want to differentiate between objects which happens to have the same set-theoretic definition, you can simply index your objects of interest to make them unique.

 

[cheesecakes]

You are considering defining ordered triples as

(a,b,c)={{a}, {a,b}, {a,b,c}}​

then? I've never seen such a definition proposed -- in fact, it is definitely inadequate because, by that definition,

(a,b,b) = (a,a,b)​

.

Oops. I goofed up.

But my point could have been phrased using other mathematical objects. And that you people have addressed . Thanks.

 

[xxxx0xxxx]

Your ordered triplet is the ordered pair: <<a,b>,c> or {{{a}.{a,b}}},{{{a}.{a,b}}},{{a}.{a,b}},{c}}}}