Set-Theoretic Notation Kuratowski

[bird34]

Hi, I am trying to work through ordered triples and quadruples, and I want to make sure I am on the right track.

Given Kuratowski's set-theoretic definition of ordered pairs: <x,y> = df {{x},{x,y}} it seems that the definition of <a, b, c> = df <<a, b>, c> would be <a, b, c> = df{{a}, {a, b}, {c}}
However, I am not certain of this and have done a ton of reading to try to guide my answer, but still haven't arrived at any definite conclusion.

Any insight would be greatly appreciated. Especially because I need to figure out how to prove that <<a, b>, c> ≠ <a, <b, c>>. Thanks!

 

[Deveno]

well, you have two different ways of defining an ordered triple:

(a,b,c) = (a,(b,c)) = {{a},{a,(b,c)}} = {{a},{a,{{b},{b,c}}}}

-OR-

(a,b,c) = ((a,b),c) = {{(a,b)}, {(a,b),c}} = {{{a},{a,b}},{{{a},{a,b}},c}}

however, there is a "natural" bijection between (AxB)xC and Ax(BxC):

define p₁(a,b) = a; p₂(a,b) = b

then show that f:(AxB)xC→Ax(BxC) given by:

f((a,b),c)) = (p₁(a,b),(p₂(a,b),c))

and g:Ax(BxC)→(AxB)xC given by:

g(a,(b,c)) = ((a,p₁(b,c)),p₂(b,c))

are inverses of each other.

to go a bit further, there is no reason why one could not DEFINE:

(a,b) = {{a,b},{b}}, it's pretty much an arbitrary choice (since we read left-to-right, "leftmost" means "first" in our minds. purely a cultural convention).

unless one wishes to include "urelements" in one's set theory, there's no way to meaningfully distinguish between sets of the same cardinality (as SETS), besides "naming" the elements differently. that is the set:

{Alice,Bob} might as well be {x,y} since the bijection:

f(Alice) = x
f(Bob) = y

just renames Alice to x and Bob to y.

or, to put it differently, often we don't care about "=" so much as we care about "~", where ~ is some suitable equivalence relation. two apples are not two oranges, but they both are "two of something".

(of course, some logicians may disagree: the set {a,b} where a is an element of A, and b is an element of B, shouldn't be "the same as" the set {A,B}. so a lot of this depends on context; that is, the alphabet of a formal language may play a role in distinguishing sets in some essential way).

or yet again: in most contexts, "=" becomes "~" in some meta-theory, where equality is defined more strictly. for example:

2+2 = 4, when 2 and 2 and 4 are integers, and + is ordinary addition
2+2 ≠ 4, when 2+2 and 4 are data strings (or formula expressions).

 

[bird34]

Thank you so much. Some of it is starting to make sense. This is all new to me, so I do have a few other questions ...

I understand why the Kuratowski definition is
(a, b, c) = ((a, b), c) = {{(a,b)}, {(a,b),c}} = {{{a},{a,b}},{{{a},{a,b}},c}}
This makes sense and it is easy to see where and how I was screwing up.

However, how would I figure out what |((a,b), c)| is then? I guess I'm just failing to grasp where the number comes from because I realize it is more than just simply counting what is contained in the sets.

Also, I'm still stumped on trying to prove <<a, b>, c> ≠ <a, <b, c>>. I get (in general) what you are saying, but don't actually know how to show those are inverses of one another.

I really appreciate all of the help!