5307: on ordered pairs
Phil
5307

Fall 2008

notes [3Sep08]
The aim of this handout is to go through a proof of Kuratowski’s approach to ordered pairs. The hope is thatthe exercise may serve to ‘ﬁrm up’ some basic settheoretic ideas and some basic proof methods. (Note that theimportance is not so much the result as the practice at doing some settheoretic proofs.)
1 The desire for order
Our account of sets treats membership as the sole criterion of identity: if
X
and
Y
are sets, then they’re the
same set
iﬀ they have the same members. Hence, the set containing Jc’s cats is the same entity as the set containingMax and Agnes, which is the same entity as the set containing Agnes and Max.Sometimes, we want to talk of
ordered sets
, entities whose identity relies not just on membership but also
order
. In particular, we sometimes want to talk about the set containing Max and Agnes
in that order
, whichis diﬀerent from the set containing Max and Agnes. (If you remember your high school geometry, think aboutCartesian coordinates, where (1
,
2) is supposed to be diﬀerent from (2
,
1). Clearly, (1
,
2) cannot be
{
1
,
2
}
if, as issupposed in Cartesian geometry, the order doesn’t matter.)
2 What we want: ordered pairs
What we want are entities wherein, like sets, membership is essential to identity but, unlike sets, it isn’t enough;we also demand that
order of entities
matter. What we want are ‘pairish’ entities, say
x,y
that are identiﬁednot only by their members (viz.,
x
and
y
) but also the order of these elements. More speciﬁcally, we want thefollowing condition met by our identities.O.
b,c
=
d,e
iﬀ
b
=
d
and
c
=
e
.Clearly,
b,c
=
{
b,c
}
. (Why?) The question is what to do.
3 One option
One option is to posit a new kind of entity called
ordered sets
(or whatever). The idea, in short, is to say thatin addition to our
sets
– which are one kind of entity – we expand our ontology to recognize a diﬀerent sort of entity called
ordered sets
. This expands our ontology since the latter sort of entity has diﬀerent – nonequivalent– identity conditions.
4 Kuratowski option
Kuratowski showed that we could leave our ontology as it was, just recognizing our sets. More clearly, he showedthat if Def 1 is what we require of ‘ordered sets’, then there are already sets that do the trick. In particular, hesuggested the following deﬁnition of ordered pairs.
Definition 1
x,y
=
df
{{
x
}
,
{
x,y
}}
1
5307 [MMVL]: intro sets
2
5 D1 does the trick: proof of O
We can prove that D1 does what we want by proving that
x,y
, so deﬁned, satisﬁes O. Here is a proof (brokenup into parts as an aid).
5.1 Proof: D1 entities satisfy O
What we want to prove is
{{
b
}
,
{
b,d
}}
=
{{
c
}
,
{
c,e
}}
iﬀ
b
=
c
and
d
=
e
Since this is a biconditional, we break it into two directions (the two component conditionals).
5.1.1 RLD
Assume, for conditional proof, that
b
=
c
and
d
=
e
. We need to show that
{{
b
}
,
{
b,d
}}
=
{{
c
}
,
{
c,e
}}
. But thisfollows immediately from our assumption and laws (substitution) of identity.
5.1.2 LRD
For conditional proof, we assume that
{{
b
}
,
{
b,d
}}
=
{{
c
}
,
{
c,e
}}
, and need to show that
b
=
c
and
d
=
e
. By thecriterion of identity for sets (viz., Extensionality),
{{
b
}
,
{
b,d
}}
=
{{
c
}
,
{
c,e
}}
iﬀ the sets have the same members.Now,
{{
b
}
,
{
b,d
}}
=
{{
c
}
,
{
c,e
}}
have the same members iﬀ either C1 holds or C2 holds.C1.
{
b
}
=
{
c
}
and
{
b,d
}
=
{
c,e
}
.C2.
{
b
}
=
{
c,e
}
and
{
b,d
}
=
{
c
}
.What we need to show is that the consequent of the LRD (viz., that
b
=
c
and
d
=
e
) holds in each case. Theseare taken in turn. In what follows, ‘Ext’ abbreviates ‘Extensionality’.
ã
Case C1.
By Ext,
{
b
}
=
{
c
}
iﬀ
b
=
c
. Hence, by Ext,
{
b,d
}
=
{
c,e
}
only if
d
=
e
. Hence, if C1 holds then
b
=
c
and
d
=
e
.
ã
By Ext, if
{
b
}
=
{
c,e
}
then
b
=
c
=
e
, and if
{
b,d
}
=
{
c
}
then
b
=
d
=
c
. But, then, by transitivity of identity,
1
d
=
e
. Hence, if C2 holds, then
b
=
c
and
d
=
e
.
6 Generalizing...
The generalization, roughly put, takes ordered
n
tuples (for
n
≥
2) to be ordered pairs along the following pattern.
ã
triples:
x,y,z
=
x,
y,z
ã
quadruples:
w,x,y,z
=
w,
x,y,z
ã
etcIt may be useful, as an exercise, to ‘cash out’ triples into primitive notation – and then show that they satisfy theD1 principle. (Consider this to be extra homework, though this needn’t be done before class on 3 September.)
1
Transitivity of Identity has it that if
x
=
y
and
y
=
z
then
x
=
z
for all
x,y,z
.