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5307: on ordered pairs

Phil 5307 | Fall 2008 | notes [3Sep-08]

The aim of this handout is to go through a proof of Kuratowskis approach to ordered pairs. The hope is thatthe exercise may serve to firm up some basic set-theoretic ideas and some basic proof methods. (Note that theimportance is not so much the result as the practice at doing some set-theoretic proofs.)

1 The desire for order

Our account of sets treats membership as the sole criterion of identity: ifX andYare sets, then theyre thesamesetiff they have the same members. Hence, the set containing Jcs 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 ofordered sets, entities whose identity relies not just on membership but alsoorder. 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 doesnt matter.)

2 What we want: ordered pairs

What we want are entities wherein, like sets, membership is essential to identity but, unlike sets, it isnt enough;we also demand that order of entitiesmatter. What we want are pair-ish 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 iffb= 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 ofentity called ordered sets. This expands our ontology since the latter sort of entity has different non-equivalent 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}}

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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 this

follows 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.

Case C1. By Ext, {b}={c}iffb= c. Hence, by Ext, {b, d}= {c, e}only ifd= e. Hence, if C1 holds thenb= c and d= e.

By Ext, if{b} = {c, e} then b = c = e, and if{b, d} = {c} then b = d = c. But, then, by transitivity ofidentity,1 d= e. Hence, if C2 holds, then b = c and d= e.

6 Generalizing. . .

The generalization, roughly put, takes orderedn-tuples (forn 2) to be ordered pairs along the following pattern.

triples: x,y,z= x, y, z

quadruples: w,x,y,z=w, x,y,z

etc

It 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 neednt be done before class on 3 September.)

1Transitivity of Identity has it that if x= y and y = z then x= z for all x,y, z.